Polarization state generation with a metasurface

ABSTRACT

The present disclosure provides an optical component, which may be a metasurface grating, including (a) a substrate; and (b) an array of subwavelength-spaced phase-shifting elements, which are tessellated on the substrate to produce, when illuminated with a polarized incident light, a diffracted light beam with a distinct polarization state for each of a finite number of diffraction orders, wherein the finite number is 2 or more.

CROSS REFERENCE TO RELATED APPLICATION

This application is a continuation of U.S. patent application Ser. No.16/964,058, filed Jul. 22, 2020, which is a National Stage Entry ofInternational Application No. PCT/US2019/014975, filed Jan. 24, 2019,which claims the benefit of and priority to U.S. Provisional PatentApplication No. 62/621,453, filed on Jan. 24, 2018, the contents ofwhich are incorporated herein by reference in their entirety.

STATEMENT OF FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT

This invention was made with Government support under the NationalScience Foundation grants nos. DGE1144152 and Air Force Office ofScientific Research (MURI, Grants No. FA9550-14-1-0389 and No.FA9550-16-1-0156). The Government has certain rights in the invention.

FIELD

The present disclosure relates to the field of optics and moreparticularly, to polarization state generation with a metasurface.

SUMMARY

One embodiment provides for an optical component including (a) asubstrate; and (b) an array of subwavelength-spaced phase-shiftingelements, which are tessellated on the substrate to produce, whenilluminated with a polarized incident light, a diffracted light beamwith a distinct polarization state for each of a finite number ofdiffraction orders, wherein the finite number is 2 or more.

Another embodiment provides for an optical instrument including (A) anoptical component including (a) a substrate; and (b) an array ofsubwavelength-spaced phase-shifting elements, which are tessellated onthe substrate to produce, when illuminated with a polarized incidentlight, a diffracted light beam with a distinct polarization state foreach of a finite number of diffraction orders, wherein the finite numberis 2 or more and (B) one or more detecting elements, each configured todetect the diffracted light beam for one of the diffraction orders.

Yet another embodiment provides for a polarization testing methodincluding illuminating the above optical component with a test light;and measuring a light intensity of a beam diffracted from the opticalcomponent for each of the finite number of the diffraction orders.

BRIEF DESCRIPTION OF THE FIGURES

FIG. 1 . Polarimetry amounts to several projective measurements of anincident Stokes vector,

${\overset{\rightarrow}{S}}_{inc},$onto a number of analysis Stokes vectors

$\left\{ {\overset{\rightarrow}{S}}_{n} \right\}.$If the analysis vectors are known and linearly independent,

${\overset{\rightarrow}{S}}_{inc}$may be recovered

FIG. 2 .: a, exemplary schematic: a metasurface diffraction grating maybe designed to produce arbitrarily specified polarization states on itsdiffraction orders. The same device may also act as a parallelpolarimeter. b, such a metasurface may be composed of pillar-likephase-shifting elements with two perpendicular mirror symmetry axes(e.g., rectangles) whose orthogonal dimensions w_(x) and w_(y) may beadjusted to allow for independent and tunable phase delays Φ_(x) andΦ_(y) on x and y polarized light. c, if N such elements of varyingdimensions are arranged into a periodic unit cell along the {tilde over(x)} direction, a 1D metasurface diffraction grating may be formed. Ateach point in the unit cell, constant phases are imparted on x and ypolarized light. Then described are the phase profiles experienced bythese polarizations in the form of N-vectors Φ_(x) and Φ_(y). d, Eachdiffraction order of the grating is corresponding to a bulk opticcascade of a diattenuator and a phase retarder, each oriented along x/y.These elements enact the polarization transformations producing somepolarization on the order when a given polarization is incident. e, thePoincare sphere may aid in understanding the behavior of the diffractionorder for general input polarization. A standard Poincaré sphere (upperleft) represents the set of all possible incident polarizations. Afterpassing through the diattenuator, the sphere is distorted along the S₁axis to a degree dependent on the extinction ratio of the diattenuator(upper right). Finally, the phase retarder enacts a precession of thesphere along the S₁ axis by an angle equal to its retardance δ^((m))(bottom). On each sphere, the red arrow and blue dot denote thepolarization ellipses depicted in (d). The power of the output beam ispolarization dependent (not shown). f, in general, the functionalitycontained in a single metasurface (c) would involve a half- andquarter-waveplate on each order, that is, 2P birefringent plates, inaddition to a grating.

FIG. 3 .: a, exemplary schematic of optimization routine used to designmetasurface polarization gratings. An initial guess for the phaseprofiles {right arrow over (Φ)}_(x) and {right arrow over (Φ)}_(y) isoptimized to direct as much light as possible into the diffractionorders of interest using gradient descent under the constraints of thedesired polarization states. This guess may be improved by agradient-free method that accounts for simulated properties of the phaseshifters used, and a final geometry is generated. These geometries arerealized in TiO₂ for operation at λ=532 nm. b, the scheme in (a) is usedto generate two gratings, one for four polarizations of general interest(top) and one for a tetrahedron configuration of polarization states(bottom). Each grating generates four polarization states, and thetarget ellipse, expectation from FDTD simulation, and experimentallyobserved polarization ellipse on each grating order are shown. c, design(black) and electron micrographs as-fabricated of the “fourpolarization” (top) and “tetrahedron” (bottom) gratings. In thetetrahedron grating the exceedingly small pillar has not survivedfabrication. d, a representation of the results in (b) on the Poincarésphere. Dashed lines represent the designed polarizations and the solidshapes denote experimental measurements.

FIG. 4 .: a, as each diffraction order may be thought of as a cascade ofa diattenuator and a retarder (FIG. 2 d ), when light from a source ofknown polarization (in this case, linearly polarized at 45°) isincident, a characteristic polarization {right arrow over (S)}_(c) isproduced. If light of unknown polarization {right arrow over (S)}_(inc)is incident in the reverse direction and if the source is replaced witha detector, the measured intensity I∝{right arrow over (S)}_(inc)·{rightarrow over (S)}_(c) is obtained by time-reversal symmetry. b, this factallows the meta-grating to function as a parallel polarimeter. Each ofthe four diffraction orders of the tetrahedron grating may be used as aseparate analyzer. Light incident on the metasurface passes through alinear polarizer at 45° and diffracts onto four photodiodes whosephotocurrents are amplified and digitized through an analog-to-digitalconverter (ADC). For testing and calibration purposes, light passesthrough various polarization optics in front of the meta-grating. Therole of the boxed components (i) and (ii) are described in the text. c,As the linear polarizer is rotated in front of a polarizationMach-Zehnder interferometer ((i) in (b)) whose path length difference islarger than the laser coherence length L_(coh), the degree ofpolarization (DOP) varies. Plotted is the DOP reported by themeta-grating polarimeter which nearly identically follows thetheoretically expected curve. At 45° (inset), when the light ismaximally decohered, a DOP of p=0.2±0.176% is measured.

FIG. 5 .: In each column, the metasurface grating polarimeter(metasurface) and the commercial rotating waveplate polarimeter (RWP)are compared using different polarimetric quantities. In the top row ofgraphs, values reported by each polarimeter are plotted against oneanother (in the case of perfect correspondence all values would liealong the 1:1 line). Insets of each plot are shown. Error bars are givenfor the metasurface values since precision is not known for thecommercial RWP. In the bottom row of plots, the differences between thevalues reported by each polarimeter are computed and plotted in ahistogram. Each distribution is fitted with a normal distribution andthe mean μ and variance σ are given for each. The quantities examinedare the degree of polarization (DOP), the azimuth double angle 2θ, andthe ellipticity double angle 2∈. The latter two are parameters of thepolarization ellipse that give the spherical coordinates of thepolarization state on the Poincaré sphere.

FIG. 6 .: Evaluation of the merit criterion over 64 iterations. Shownhere is the sum merit criterion, e.g. the amount of each of x- andy-polarized light that are directed into the orders of interest. Sincein this case the incident polarization is 45°, η_(x)=η_(y). Theefficiency converges to a peak value of η_(sum)=1.4639, meaning that theoverall efficiency

$\eta = {\frac{n_{sum}}{2} = {7{3.1}{\%.}}}$

FIG. 7 .: Convergence of constraints results for the four-polarizationgrating in the text over 64 iterations.

FIG. 8 .: Polarization ellipses obtained on the four diffraction ordersfrom direct Fourier transform of the optimized phase profilesϕ_(x)({tilde over (x)}) and ϕ_(y)({tilde over (x)}).

FIG. 9 .: Schematic of an FDTD simulation of a designed grating inLumerical FDTD®.

FIG. 10 .: Polarization ellipses on diffraction orders predicted fromFDTD simulation of designed diffraction grating.

FIG. 11 .: Effect of changing nominal fabrication CAD given to e-beamsystem on polarization ellipses produced on the m=−2, −1, +1, and +2 onthe four-polarization grating.

FIG. 12 .: Effect of changing nominal fabrication CAD given to e-beamsystem on polarization ellipses produced on the m=−2, −1, +1, and +2 onthe tetrahedron grating.

FIG. 13 .: Schematic of the optical setup used during the first stage ofthe calibration of the metasurface grating polarimeter.

FIG. 14 .: Visualization: As the linear polarizer is rotated, theincident laser beam traces out a circular path on the metasurface.

FIG. 15 .: Schematic of the optical setup used during the second stageof the calibration of the metasurface grating polarimeter.

FIG. 16 .: Calibration data obtained from the exposing the polarimeterto RCP and LCP light. Averaging the values obtained on each channelyields {right arrow over (I)}_(RCP) and {right arrow over (I)}_(LCP).Note that for one circular polarization, Photodiode 1 is nearlyextinguished, while for the other it is maximal, as would be expectedfrom the designed polarization state.

FIG. 17 .: The DOP of the linear polarization states used duringcalibration, as determined with the final instrument matrix A. The factthat these DOPs never differ by more than about 0.5% indicates that thecalibration was self-consistent and that the resultant instrument matrixA can be trusted.

FIG. 18 .: Schematic of the setup used to produce and quantify partiallypolarized light with the metasurface-grating polarimeter.

FIG. 19 .: Full dataset presented for partially polarized light. Datapoints are taken every 0.25° and are shown, partially obscuring thetheoretical fit.

FIG. 20 .: A smattering of unusual results obtained in differentiterations of the same experiment. We discuss possible causes of thesetrends.

FIG. 21 .: Two beams from the polarization beamsplitter interferometeremerge having diverged by different amounts. These beams (shown in a andb) are orthogonally linearly polarized and have no phase memory of oneanother. When these beams—of equal overall intensity—overlap, the DOPvaries across the beam profile, rather than being 0 everywhere asintended. At the points where their intensities are equal, the DOP is 0,and it is nearly 1 at points where one beam dominates. Note here thatthe two beams are of the same wavelength, and that color is used herefor illustrative purpose. The difference in the sizes of the beams isexaggerated here.

FIG. 22 .: The 4×4 instrument matrix. The elements in the leftmost threecolumns are determined from the linear polarizer-only portion of thecalibration and are enclosed in solid boxes. The fourth column isdetermined from the linear polarizer plus quarter-waveplate portion ofthe calibration and is enclosed in a dotted box. We can obtain estimatesfor covariances between elements in the same box. We cannot easilyestimate the covariance for elements between boxes. Thus, we make theassumption that the 16×16 covariance matrix of A is diagonal.

FIG. 23 .: Setup used for comparison of degree of polarizationmeasurements (DOP) between polarimeters.

FIG. 24 .: Setup used for comparison of azimuth and ellipticity betweenpolarimeters.

FIG. 25 .: Full results for all polarimetric parameters obtained usingthe setup in FIG. 23 , which includes a polarization Mach-Zehnderinterferometer for the generation of partial polarization states. Theportion of this figure included in FIG. 4 a is outlined with a dottedline.

FIG. 26 .: Full results for all polarimetric parameters obtained usingthe setup in FIG. 19 , which has a LP and QWP but no Mach-Zehnderinterferometer. The portion of this figure included in FIG. 4 a isoutlined with a dotted line. Note that here, due to the absence of theMach-Zehnder interferometer, all the measured DOPs are clustered around1.0 (as expected). The standard deviations of the differences in azimuthand ellipticity in this case are about half the size as in FIG. 25 .

FIG. 27 .: Experiment to characterize angle-dependence ofpolarization-state production. A laser, aligned to the table with twomirrors, is incident on a linear polarizer at 45°. The light impinges onthe meta-grating which is on a mount so that it can be rotated by anangle of θ. A rotating waveplate polarimeter (RWP) is placed in front ofeach order in succession and the polarization data is recorded.

FIG. 28 .: Incident-angle dependent study for meta-grating designed togenerate a tetrahedron of Stokes vectors. In (a), for each diffractionorder, the three Stokes parameters {S₁, S₂, S₃} that dictate thepolarization ellipse are plotted as a function of θ. In (b), theresultant polarization ellipses are for all incident angles studied.

FIG. 29 .: Incident-angle dependent study for meta-grating designed togenerate +45°RCP/LCP/−45°. In (a), for each diffraction order, the threeStokes parameters {S₁, S₂, S₃} that dictate the polarization ellipse areplotted as a function of θ. In (b), the resultant polarization ellipsesare for all incident angles studied.

FIG. 30 .: This plot shows the same results as FIGS. 28 and 29 in thePoincare sphere representation. The more transparent the point, thehigher the angle of incidence.

FIG. 31 .: A schematic of the procedure used to analyze the effect oferrors in the angle of incidence, or more generally of any effect thatcontributes to error in the instrument matrix, on reported Stokesvectors from the polarimeter.

FIG. 32 a-c: Results of angle-dependent polarimetry study. Note that DOPerrors are expressed in absolute terms (e.g., not in %).

FIG. 33 a-d: A plot of −x and −y (top) and tx and ty (bottom), theamplitude transmission at each position. It can be seen that the libraryof structures used does not afford sufficient freedom to realize thedesired, optimized phases for x- and y-polarizations at each point withhighly uniform amplitude transmission.

FIG. 34 : Design of a grating controlling six diffraction orders'polarization states simultaneously. The aim was to produce LCP, x, 135,45, y, and RCP light on the innermost six diffraction orders. In the toprow, polarization ellipses from the result of a purely mathematical,Fourier transform phase profile optimization are shown. In the bottomrow, simulation results of a grating designed using two-stepoptimization described above (including the pattern search step). Thepolarization ellipses are notably distorted.

DETAILED DESCRIPTION

Unless otherwise specified, “a” or “an” refers to one or more.

As used herein, the terms “approximately,” “substantially,”“substantial” and “about” are used to describe and account for smallvariations. When used in conjunction with an event or circumstance, theterms can refer to instances in which the event or circumstance occursprecisely as well as instances in which the event or circumstance occursto a close approximation. For example, when used in conjunction with anumerical value, the terms can refer to a range of variation less thanor equal to ±10% of that numerical value, such as less than or equal to±5%, less than or equal to ±4%, less than or equal to ±3%, less than orequal to ±2%, less than or equal to ±1%, less than or equal to ±0.5%,less than or equal to ±0.1%, or less than or equal to ±0.05%. Forexample, two numerical values can be deemed to be “substantially” thesame if a difference between the values is less than or equal to ±10% ofan average of the values, such as less than or equal to ±5%, less thanor equal to ±4%, less than or equal to ±3%, less than or equal to ±2%,less than or equal to ±1%, less than or equal to ±0.5%, less than orequal to ±0.1%, or less than or equal to ±0.05%.

Additionally, amounts, ratios, and other numerical values are sometimespresented herein in a range format. It is to be understood that suchrange format is used for convenience and brevity and should beunderstood flexibly to include numerical values explicitly specified aslimits of a range, but also to include all individual numerical valuesor sub-ranges encompassed within that range as if each numerical valueand sub-range is explicitly specified.

An optical component is developed, which will also referred to as ametasurface grating, such that when illuminated with an incident lightbeam with a known polarization, it produces a diffraction beam with adistinct polarization state for each of a finite number (or set) ofdiffraction orders, which number is at least 2.

The optical component may include a substrate and an array ofphase-shifting elements positioned on the substrate in a specificmanner. The array may be a one-dimensional array or a two-dimensionalarray. Each of the phase shifting element may have each of its lateraldimensions, e.g. its dimensions parallel to a surface of the substrate,having subwavelength values, e.g. values no greater or smaller than awavelength of the incident light. A dimension of the phase shiftingperpendicular to the substrate's surface may be at least the same orgreater than a wavelength of the incident light. A number ofphase-shifting elements in the array may vary. In some embodiments, thenumber of phase-shifting elements may be from 8 to 100 or from 10 to 60or from 10 to 40 or any integer number or subrange within these ranges.Preferably, a lateral spacing between individual phase-shifting elementshas subwavelength values, e.g. is no greater or smaller than awavelength of the incident light.

The substrate may be made of a number of materials. In some embodiments,the substrate may be a glass substrate or a quartz substrate.

The substrate may have relatively small lateral dimensions. In someembodiments, each lateral dimension of the substrate may be no more than2 mm or no more than 1.5 mm or no more than 1 mm or no more than 0.75 mmor no more than 0.5 mm or no more than 0.4 mm or no more than 0.3 mm orno more than 0.2 mm or no more than 0.15 mm or no more than 0.1 mm.

Phase-shifting elements are made of a material, which provides asufficiently strong contrast with a surrounding medium, such as air, ata particular wavelength, while not absorbing a light at that length.Thus, a selection of a material may depend on a particular wavelengthvalue or range, at which the optical element will be used. Thesufficiently strong contrast at a particular wavelength may mean that amaterial of phase-shifting elements has a refractive index valuesignificantly greater than that of a surrounding medium, such as air.Thus, a material of phase-shifting elements may have a refractive indexfor a particular wavelength of at least 2.0, or at least 2.1 or at least2.2 or at least 2.3 or at least 2.4 or at least 2.5 or at least 2.6 orat least 2.7 or at least 2.8 or at least 2.9 or at least 3.0 or at least3.1 or at least 3.2 or at least 3.3 or least 3.4 or at least 3.5.Preferably, a material of phase-sifting elements does not absorb at awavelength of the incident light.

In some embodiments, the phase-shifting elements may comprise titaniumdioxide, silicon nitride, an oxide, a nitride, a sulfide, a pureelement, or a combination of two or more of these.

In some embodiments, the phase-shifting elements may comprise metal ornon-metal oxides, such as, alumina (e.g. Al₂O₃), silica (e.g. SiO₂),hafnium oxide (e.g. HfO₂), zinc oxide (e.g. ZnO), magnesium oxide (e.g.MgO), titania (e.g. TiO₂), metal or non-metal nitrides, such as nitridesof silicon (e.g. Si₃N₄), boron (e.g. BN) or tungsten (e.g. WN), metal ornon-metal sulfides, pure elements (e.g. Si or Ge, which may be used forlonger wavelengths, such near IR or mid-IR wavelengths).

Certain comparative methods of fabricating metasurfaces are described inR. C. Devlin, et al., Proc. Natl. Acad. Sci. 113, 10473 (2016).

The number of diffraction orders may be any preselected number. Forexample, the number of diffraction orders may be any integer from 2 to20 or from 2 to 12 or from 2 to 8. Optical component with a smallernumber of diffraction orders, such 2, 3, 4, 5 or 6, may have moreapplications.

The phase shifting elements are configured (e.g., positioned on thesubstrate) such that when illuminated with an incident polarized light,light intensities for each of the pre-selected diffraction orders (thefinite number of diffraction orders) are approximately equal to eachother, while light intensities for any other possible diffraction ordersare much smaller, preferably at least one or at least orders ofmagnitude less than light intensities for the preselected diffractionorders, and more preferably below a limit of detection for a detectingelement.

In some embodiments, the phase shifting elements are configured toproduce four distinct polarization states.

For example, in some embodiments, the phase shifting elements areconfigured so that when the phase shifting elements are illuminated witha +45° linear polarized (relative to the surface of the substrate)light, the grating produces diffraction beams at −2, −1, +1 and +2diffraction orders with a +45° linear polarized state, a right circularpolarized state, a left circular polarized state and a −45° linearpolarized state, respectively.

In some embodiments, the phase-shifting elements are configured toproduce at −2, −1, +1 and +2 diffraction orders four polarization statescorresponding to vertices of a tetrahedron inscribed in the Poincaresphere when illuminated with an incident light which is +45° linearpolarized relative to a surface of the substrate.

In some embodiments, polarization states for different diffractionorders may be linearly independent.

Yet in some embodiments, two or more of polarization states fordifferent diffraction orders may be linearly dependent.

The optical component (metasurface grating) may be used in an opticalinstrument, which may further comprise one or more detecting elementseach configured to detect a diffracted light beam for one of thediffracting orders of the grating. The optical instrument may be apolarimeter.

Due to the finite number of diffraction orders, the optical instrumentmay include a finite number of detecting elements, which may correspondto the number of the diffraction orders.

In some embodiments, a detecting element used in the optical instrumentmay be a single wave detector, e.g. a detector configured to measure anintensity of light at a particular single wavelength. In someembodiments, the single wavelength detector may be a detector, which hasa linear response to a light intensity at the wavelength of the incidentlight. The single wave length detector may be a DC detector or an ACdetector.

In some embodiments, a detecting element in the optical instrument maybe a multi-wavelength detector, e.g. a detector configured to measure anintensity of light in a range of wavelengths. The optical instrumentequipped with multi-wavelength detectors may function as a spectroscopicpolarimeter.

In some embodiments, a detecting elements used in the optical instrumentmay be an imaging sensor. The optical instrument equipped with imagingsensors may function as a polarization imaging instrument.

Preferably, the optical instrument does not include any birefingentoptical element.

Preferably, the metasurface grating is the only metasurface component ofthe optical instrument, e.g. the optical instrument does not include anymetasurface component other than the metasurface grating.

In some embodiments, the optical instrument may comprise a firstpolarizer positioned on an optical path of an incident test lighttowards the optical component. The first polarizer may have anextinction coefficient between 1000 and 200000 or between 5000 and150000 or between 5000 and 120000 or any value or subrange within theseranges.

In some embodiments, the optical instrument may comprise a secondpolarizer positioned on an optical path of diffracted beam(s) towardsthe detecting elements. The second polarizer may have an extinctioncoefficient between 500 and 200000 or between 1000 and 150000 or between5000 and 120000 or any value or subrange within these ranges. In certainembodiments, the second polarizer may have a lower extinctioncoefficient, such as between 500 and 20000 or between 5000 and 12000 orbetween 600 and 8000 or between 600 and 5000 or between 600 and 4000 orbetween 600 and 3000 or any value or subrange within these ranges.

In some embodiments, the optical instrument may comprise a lenspositioned on an optical path of the diffracted beam(s) towards thedetecting element(s). The lens may allow reducing the size of theoptical instrument.

The metasurface grating may be used for testing a polarization of a testlight with unknown polarization. For example, the metasurface gratingmay be illuminated with the test light; and then a light intensity of abeam diffracted from the metasurface grating may be measured for each ofthe finite number of the diffraction orders. In some embodiments, thetest light may be a partially polarized or an unpolarized light.

In some embodiments, when the metasurface grating is used in apolarimeter, the grating may be calibrated for a particular incidentangle of a calibration incident polarized light. In some embodiments,the metasurface grating may be used in a polarimeter for testing apolarization of a test light with unknown polarization even when anincident angle of the test light slightly differs from the calibrationincident angle of the calibration incident polarized light. For example,in some embodiments, the incident angle of the test light may be within±7° or ±6° or ±5° or ±4° of the calibration incident angle of thecalibration incident polarized light.

Embodiments described herein are further illustrated by, though in noway limited to, the following working examples.

Working Examples

Polarization State Generation and Measurement in Parallel with a SingleMetasurface

The constituent elements of metasurfaces may be designed with explicitpolarization-dependence, making metasurfaces a platform for newpolarization optics. This disclosure shows that a metasurface gratingcan be designed producing arbitrarily specified polarization states on aset of defined diffraction orders given that the polarization of theincident beam is known. Also demonstrated is that, when used in areverse configuration, the same grating may be used as a parallelsnapshot polarimeter, including a minimum of bulk polarization optics.This disclosure demonstrates its use in measuring partially polarizedlight, and shows that it performs favorably in comparison to acommercial polarimeter. The results of this disclosure may be used in anumber of applications, which may involve lightweight, compact, and lowcost polarization optics, polarimetry, or polarization imaging.

INTRODUCTION

Polarization holds a role of importance in countless areas of scienceand technology, in areas as diverse as atomic physics and fundamentallight/matter interaction, to fiber-optic telecommunications andpolarization-resolved imaging. The latter has found application inremote sensing, aerosol characterization, non-invasive cancer pathology,and astrophysics. Methods of producing, measuring, and manipulatingpolarized light, then, are of significant scientific and technologicalinterest. The basic units of polarization optics in free space usuallyinclude polarizers and/or phase retarders (waveplates). Variouspolarizer technologies exist, including wire-grids, dichroic crystals,birefringent crystal prisms, and polarizing sheets. Phase retarders arecommonly formed of bulk bi/uniaxial crystals. Their birefringentproperties allow for polarization conversion and led to the originaldiscovery of light's polarization. These plates, however, may bedifficult to fabricate and/or process and challenging to integrate,especially with minituarized optics.

The measurement of polarization is usually referred to as polarimetry[12]. Stokes polarimetry in particular refers to the determination ofthe full, four-component polarization Stokes vector {right arrow over(S)}=[S₀ S₁ S₂ S₃]^(T), which quantifies the shape, orientation,intensity, and degree of polarization of the polarization ellipse.Polarization generation and analysis are conjugate; any configuration ofpolarization optics serving as a polarization state generator may be ananalyzer, if used in reverse. If an unknown Stokes vector {right arrowover (S)}_(inc) is incident on an analyzer, a detector would observe, asa consequence of this symmetry, I_(meas)∝{right arrow over(S)}_(inc){right arrow over (S)}_(c), where {right arrow over (S)}_(c)is the characteristic polarization produced by the analyzer, were itused as a generator. Polarimetry amounts to several such projectivemeasurements of the Stokes vector (FIG. 1 ). This may be formalized inthe matrix equation

$\begin{matrix}{{A\mspace{14mu}\overset{\rightarrow}{S}} = \overset{\rightarrow}{I}} & (1)\end{matrix}$

A is an N×4 matrix known as the instrument matrix, {right arrow over(S)}_(inc) is an incident Stokes vector, and {right arrow over (I)} is alist of N measured intensities. A links the parameters of the Stokesvector with N measured intensities {right arrow over (I)} on N analyzerchannels. In the special case where N=4 the Stokes vector can bedirectly written as {right arrow over (S)}_(inc)=A⁻¹{right arrow over(I)}(in the over-determined case where N>4, one finds a least-squaressolution for {right arrow over (S)}_(inc)) [12].

Several broad categories of Stokes polarimeters exist which vary in howthe N desired projective measurements are implemented. In thedivision-of-time approach, N measurements are taken sequentially in timeas a configuration of polarization optics changes. While this reducesthe number of necessary components, time resolution is limited by thespeed at which the polarization optics may be readjusted. In the case ofmechanical rotation, this may represent a severe handicap. Activepolarization optics, such as liquid-crystal variable retarders, mayameliorate this issue somewhat, though here too, time resolution islimited to the ms range, at great expense [5]. In thedivision-of-amplitude (also known as parallel, or snapshot, polarimetry)approach, on the other hand, the wavefront is divided among N parallelchannels each of which contains a different analyzer. This may beaccomplished, for example, by the use of birefringent (e.g., Wollaston)prisms and beamsplitters [13], or by employing a diffraction grating tosplit the beam into N orders containing unique polarization optics and adetector [14, 15]. Division-of-amplitude may be desirable because thetime-resolution of polarization determination may be limited only by thedetection electronics. A major drawback of the division-of-amplitudeapproach, however, may be the demand for distinct polarization optics oneach channel, which increases complexity and bulk.

The basic units of these polarization optics (in free space, at least)may be polarizers and phase retarders (waveplates). Retarders arecommonly formed from bulk bi/uniaxial crystals whose birefringentproperties allow for polarization conversion; these led to the originaldiscovery of light's polarization. These plates are, however, difficultto fabricate and process and challenging to integrate [2, 16].

Meanwhile, metasurfaces [17]—which may be defined assubwavelength-spaced arrays of nanophotonic phase-shifting elements—haveattracted significant interest and may hold promise for miniaturizationof a variety of bulk optics. The elements of a metasurface may possesstailored structural birefringence [18, 19].

This disclosure presents a scheme for designing of a metasurface gratingthat, when light of a known polarization is incident, may producearbitrarily specified states of polarization in parallel on itsdiffraction orders (FIG. 2 a ). This disclosure experimentallycharacterizes two gratings designed with this scheme. The same grating,by the symmetry described above, may act as a parallel full-Stokespolarimeter requiring no bulk birefringent optics. This disclosurecharacterizes such a polarimeter and compares its performance to acommercial rotating-waveplate instrument. The results of this disclosuremay be used in a number of applications, which may involve lightweight,compact, and low cost polarization optics, polarimetry, or polarizationimaging.

Principle of Operation

A subwavelength metasurface element with two perpendicular symmetry axes[20] (e.g., a rectangle, but other conceivable examples are encompassed)can function as a waveplate-like phase shifter, imparting independentphase shifts Φ_(x) and Φ_(y) on x and y polarized light [18, 19]. Thevalues of Φ_(x) and Φ_(y) may be arbitrarily adjusted between 0 and 2πby changing the perpendicular dimensions w_(x) and w_(y) (FIG. 2 b ). IfQ such birefringent phase shifters are arranged with subwavelengthspacing in a 1D periodic grating unit cell (FIG. 2 c ), the phase shiftexperienced by x polarized light at the q^(th) position in the unit cellcan be denoted by ϕ_(x) ^((q)). That is, approximation of the phaseshift acquired by the wavefront at each position in the unit cell can bemade as constant. This discrete phase function Φ_(x)(

experienced by x polarized light, as a function of the spatialcoordinate {tilde over (x)} (not to be confused with x polarized light),can be written as a vector, {right arrow over (ϕ)}_(x)={ϕ_(x) ⁽¹⁾, . . .,

} with {right arrow over (ϕ)}_(y) holding an analogous meaning for ypolarized light. If the periodic unit cell is tessellated, a metasurfacephase grating is formed that implements independent and arbitraryperiodic phase profiles for orthogonal x and y polarizations.

Being periodic, the grating's angular spectrum is discrete. Given thephase profiles ϕ_(x)({tilde over (x)}) and ϕ_(y)({tilde over (x)})(which are contained in {right arrow over (ϕ)}_(x) and {right arrow over(ϕ)}_(y), the Fourier decomposition can be computed of each phasegrating onto the m^(th) grating order, given by

${c_{x}^{(m)} = {\left\langle {m❘e^{i^{\phi}{x{(\overset{\sim}{x})}}}} \right\rangle = {\frac{1}{2\pi}{\int_{0}^{d}{e^{i^{\phi}{x{(\overset{\sim}{x})}}}e^{i\frac{2\pi\; m\overset{\sim}{x}}{d}}d\overset{\sim}{x}}}}}}\mspace{14mu}$and$c_{y}^{(m)} = {\left\langle m \middle| e^{i^{\phi}{y{(\overset{\sim}{x})}}} \right\rangle = {\frac{1}{2\pi}{\int_{0}^{d}{e^{i^{\phi}{y{(\overset{\sim}{x})}}}e^{i\frac{2\pi\; m\overset{\sim}{x}}{d}}d\overset{\sim}{x}}}}}$where d is the length of the periodic unit cell and {c_(x) ^((m))}{c_(y) ^((m))} are the Fourier coefficients of the gratings experiencedby x and y polarizations, respectively.

Each coefficient is in general complex, so we may write c_(x)^((m))=|c_(x) ^((m))|e_(x) ^(iδ(m)) and c_(y) ^((m))=|c_(y) ^((m))|e_(y)^(iδ(m)). Then, one can ascribe to each order a Jones matrix J^((m)):

${J(m)} = {\begin{bmatrix}c_{x}^{(m)} & 0 \\0 & c_{y}^{(m)}\end{bmatrix} = {\begin{bmatrix}{c_{x}^{(m)}} & 0 \\0 & {c_{y}^{(m)}}\end{bmatrix}\begin{bmatrix}e_{x}^{i{\delta{(m)}}} & 0 \\0 & e_{y}^{i{\delta{(m)}}}\end{bmatrix}}}$

The polarization properties of order m contained in J^((m)) may be seenas corresponding to a cascade of two bulk optical elements (FIG. 2 d ):the first Jones matrix in the product is that of a diattenuator—that is,an imperfect polarizing element selectively attenuating light along thex and y directions, while the second Jones matrix is that of a phaseretarder—a waveplate—with retardance δ^((m))=δ_(x) ^((m))−δ_(y) ^((m)).Both have their eigenaxes mutually oriented along x and y (FIG. 2 d ).

If, for instance, a beam linearly polarized at 45° with electric fieldamplitude E₀ is incident on the grating, the electric field on them^(th) grating order will be:

${\overset{\rightarrow}{E}}^{(m)} = {\frac{E_{0}}{\sqrt{2}}\begin{bmatrix}c_{x}^{(m)} \\c_{y}^{(m)}\end{bmatrix}}$

In the special case of 45° polarized light, then, the complex gratingcoefficients {c_(x) ^((m))} and {c_(y) ^((m))} directly yield thepolarization state of order m. For a general input polarization, theoutput polarization state on each order can be understood with aid ofthe Poincaré sphere (FIG. 2 e , see caption).

Optimization

Given a grating with known {right arrow over (ϕ)}_(x) and {right arrowover (ϕ)}_(y) an incident beam of known polarization, the polarizationstate and power on each diffraction order m may be computed with Fourieroptics. Conversely, can one deduce the {right arrow over (ϕ)}_(x) and{right arrow over (ϕ)}_(y) that produce diffraction orders withspecified states of polarization, for a given incident polarization?This would allow for the straight-forward engineering of such gratings,embedding in a single metasurface a functionality that would otherwiseinvolve, in the most general case, an ordinary diffraction grating with2P half- and quarter-crystalline waveplates, where P is the number ofdiffraction orders to be controlled (FIG. 2 f ).

Suppose that for each diffraction order in a set {l} desired outputpolarization states can be specified. These polarizations directlydictate {c_(x) ^(m)} and {c_(y) ^(m)}, the Fourier coefficients. Thegrating could be found by simply inverting the Fourier transform. In thecase of incident light polarized at 45°, the holographic mask is givenby:

$\sum\limits_{m \in {\{ l\}}}{\left( {{c_{x}^{(m)}\ \begin{bmatrix}1 \\0\end{bmatrix}} + {c_{y}^{(m)}\ \begin{bmatrix}0 \\1\end{bmatrix}}} \right)e^{- i^{\frac{2\pi\; m\overset{\sim}{x}}{d}}}}$

However, Eq. 6, being the sum of many spatial harmonics of the grating,involves both amplitude and phase modulation. In the realm ofmetasurfaces, this may be undesirable. One may generally hope to obtaina range of phase-shifter geometries with nearly uniform amplitudetransmission that yield phase shifts ranging between 0 and 2π [18]. Itis generally difficult—at least, drawing from a finite set of possiblegeometries of streamlined design—to assemble a library of structuresyielding arbitrarily shape-tunable phase shift and transmissionsimultaneously. In the present case, it is desired that this beachievable for both x and y polarizations, simultaneously andindependent of one another. This is, without resorting to a very largerange of simulated geometries, untenable.

It is desired, then, a phase-only grating. It can be shown, however,that a phase-only grating may have one or infinitely many diffractionorders, so the exact solution (Eq. 6) is in general not phase-only [21].Thus, optimization may be necessary in order to concentrate as muchdiffracted light in the orders of interest, while taking on the desiredtarget polarization states.

More formally, it is desired to design a grating that, when lightlinearly polarized at 45° is incident, produces desired polarizationstates on a set of grating orders {

}. The target Jones vector on each order m∈{

} is given as

${\overset{\rightarrow}{J}}^{{(m)}_{=}}\begin{bmatrix}{\cos\;\mathcal{X}^{(m)}} \\{\sin\;\mathcal{X}^{(m)}e^{{\mathcal{i}\phi}^{(m)}}}\end{bmatrix}$

Light will generally be diffracted into all orders, not just those in {

}. In order to direct as much of the incident power as possible intothese desired orders, it is sought to maximize

${\eta\left( {{\overset{\rightarrow}{\Phi}}_{x},{\overset{\rightarrow}{\Phi}}_{y}} \right)} = {\sum\limits_{m \in {\{ l\}}}\sqrt{\left( {c_{x}^{(m)}\left( {\overset{\rightarrow}{\Phi}}_{x} \right)} \right)^{2} + \left( {c_{y}^{(m)}\left( {\overset{\rightarrow}{\Phi}}_{y} \right)} \right)^{2}}}$under the constraints

$\frac{c_{y}^{(m)}}{c_{x}^{(m)}} = {\tan\;\chi^{(m)}}$ andδ_(x)^((m)) − δ_(y)^((m)) = ϕ^((m))

The constraints provide for the desired polarization on each order, andthe phase profile vectors {right arrow over (Φ)}_(x) and {right arrowover (Φ)}_(y) are the quantities to be optimized. If the grating has Qconstituent elements, the optimization will involve 2Q parameters. Q andthe inter-element separation dictate the grating period d which, alongwith the operating wavelength λ, specifies the angular separation of thegrating orders. Once optimized {right arrow over (Φ)}_(x) and {rightarrow over (Φ)}_(y) are obtained, the power in the desired orders andcorrespondence with the target polarization can be mathematicallyevaluated (cf. Eqns. 8, 9, and 10).

A gradient descent optimization is performed of η({right arrow over(Φ)}_(x), {right arrow over (Φ)}_(y)) under the above constraints, withrandomly generated initial conditions (FIG. 3 a ). This may be a purelymathematical exercise and is independent of any particular materialimplementation or wavelength. Once optimized {{right arrow over(Φ)}_(x), {right arrow over (Φ)}_(y)} are found, appropriate phaseshifting geometries in the material of interest may be deduced.

For reasons pertaining to the choice of material implementation(detailed herein), a second step may be added to the optimization inwhich the results from gradient descent are improved by use of agradient-free method that explicitly uses the simulated properties ofthe TiO₂ pillar phase shifters used in this disclosure at a wavelengthof λ=532 nm (FIG. 3 a ).

Polarization State Generation

Using this two-step optimization strategy two gratings were designed foroperation at λ=532 nm, chosen owing to the scientific and technologicalubiquity of the visible range. For each element in the optimized {rightarrow over (Φ)}_(x) and {right arrow over (Φ)}_(y), a rectangular TiO₂pillar, 600 nm in height whose dimensions best impart the desired phaseson x and y polarized light is selected from a library of simulatedstructures. The designed gratings are then fabricated on a glasssubstrate.

A first grating is designed to produce +45° linear, right-circular,left-circular, and −45° linear polarizations on the m=−2, −1, +1, and +2diffraction orders, respectively, all with equalized intensities, when45° linear polarized light is incident. These represent a set ofpolarizations commonly encountered in optics experiments and are thus ofgeneral interest. This is referred to as the “four polarization”grating. A second grating is designed to produce four polarizationstates corresponding to the vertices of a tetrahedron inscribed in thePoincaré sphere on the same orders with equalized intensities for thesame incident polarization. This set of polarizations is of significancein polarimetry (discussed below) [23, 24]. This is referred to as the“tetrahedron grating”.

Both gratings contained Q=20 individual elements, so each involved theoptimization of 2Q=40 parameters. This Q was found, heuristically, toproduce results that achieve both high efficiency η and goodcorrespondence with the desired polarization ellipses—bothmathematically and from FDTD simulation—while minimizing the number ofoptimization parameters. However, larger or smaller number of elementsmay be also used.

The unit cell geometries implementing the optimized phase profiles foreach grating are shown in FIG. 3 c alongside corresponding electronmicrographs. Each unit cell was tessellated into bulk metasurfacegratings each 250×250 μm in size.

Each grating was illuminated with laser light at λ=532 nm linearlypolarized at 45° relative to the axes of the grating. The polarizationstate of the light on each of the diffraction orders of interest wasthen measured with a commercial rotating waveplate polarimeter (furtherdetails about this measurement are deferred to the supplement).

In FIG. 3 b , for each grating, the measured polarization ellipses oneach order are plotted alongside the desired target ellipses as well asthe ellipses predicted by an FDTD simulation of the grating geometry. Aqualitatively close correspondence between the desired target,simulated, and observed polarization states was observed.

Metasurface Polarimetry

Each order of the metasurface polarization grating may be thought of asa diattenuator in series with a phase retarder, each oriented along x/y(Eqn. 4 and FIG. 3 d ). When light from a source passes through apolarizer oriented at 45°, a polarization state is produced on thegrating order, ideally close to some target state (FIG. 4 a , top). Whenthe grating is used in reverse—that is, with the grating followed by alinear polarizer oriented at 45°—each diffraction order may be seen as apolarization state analyzer for its characteristic Stokes vector (FIG. 4a , bottom).

The grating may then be used as a parallel full-Stokes polarimeter withno polarization optics (with the exception of a single polarizer, whichmay be, for example, integrated on top of the grating. In someembodiments, the polarizer may be necessary for Stokes vectordetermination). This may rely on a suitable choice of analyzer states,which may be arbitrarily specified. In some embodiments, the fourpolarization grating may be not sufficient for full-Stokes polarimetryas its states in an ideal situation are not linearly independent. Insome embodiments, imperfections of the four polarization grating maybreak this and render it usable, though not ideally, for polarimetry.For a polarimeter making N=4 measurements, it has been extensivelydocumented that, in the absence of calibration errors [24], aconfiguration of analyzers whose characteristic Stokes vectorscorrespond to (any) tetrahedron inscribed in the Poincaré sphere yieldsmaximum signal-to-noise ratio in Stokes vector determination [23].

In acknowledgment of this, a larger version (1.5 mm×1.5 mm) of thetetrahedron grating described above was fabricated. When laser light isincident, the four diffraction orders of interest pass through apolarizer oriented at 45° and diverge. Some distance away (cms), eachbeam impinges on a standard silicon photodiode, producing a photocurrentwhich is amplified and converted to digital form (FIG. 4 c , rightside).

The instrument matrix A may be determined by calibration. Accordingly,this disclosure carried out a calibration scheme [25] developed for thefour-detector photopolarimeter of Azzam [26], applicable to anypolarimeter with four intensity channels (N=4) which explicitly accountsfor imperfect quarterwaveplates. The implementation of this calibrationis documented in the below disclosure. Each entry of the resultantinstrument matrix A may be assigned error bounds which provide for thefull covariance matrix of any computed Stokes vector [27], allowing oneto place uncertainty bounds on any Stokes vector predicted by themetasurface grating polarimeter.

Quantifying Partially Polarized Light

With the metasurface grating polarimeter calibrated, the Stokes vectorof any incident beam may be determined from A and the measuredintensities on the photodiodes. An interesting case may be that ofpartially polarized light. Partially and un-polarized light, inherentlya consequence of temporal coherence phenomena [2, 28], are common in allnon-laser light sources. The degree to which light is unpolarized isquantified by the degree of polarization (DOP), defined as

$p = \frac{\sqrt{S_{1}^{2} + S_{2}^{2} + S_{3}^{2}}}{S_{0}}$where S_(i) denotes the i^(th) element of the Stokes vector. Fullypolarized light corresponds to p=1, totally unpolarized light to p=0; inintermediate cases, p represents the ratio of the beam's power which ispolarized to that which is not.

In order to study the response of the metasurface-grating polarimeter tovarying DOP, a deterministic means of producing partially polarizedlight may be involved. This disclosure makes use of a Mach-Zehnder-likesetup with two polarization beamsplitters [2, 29]. This is depicted inFIG. 4 b , where the boxed components in (i) are included while (ii) isomitted (though (ii)'s presence would theoretically not affect the DOP).As a linear polarizer rotates in front of the interferometer, differentfractions of incident light are directed into each arm. When equal partsof the beam travel along each path (when θ_(LP)=45°) and the path-lengthdifference of the interferometer is arranged to be many coherencelengths L_(coh) of the laser source, the recombined beam should betotally unpolarized: The beam will be composed of half x polarized lightand half y polarized light which no longer have phase-coherence. On theother hand, when θ_(LP)=0° or 90°, the beam is completely polarizedbecause all light goes along one path. At intermediate angles, p=|cosθ_(LP)|[29].

A linear polarizer was rotated in front of the interferometer while theStokes vector reported by the meta-grating was computed. Thecorresponding DOPs (Eqn. 11) are plotted in FIG. 4 e . As shown in theinset, a minimum DOP of 1.2% is observed with an uncertainty of 0.176%.

Comparison with a Commercial Rotating-Waveplate Polarimeter

Finally, this disclosure compares the performance of themetasurfacegrating polarimeter to a commercial and widely usedvisible-range rotating waveplate polarimeter (ThorLabs modelPAX5710VIS-T). In a rotating waveplate polarimeter (RWP), a waveplatemechanically rotates in front of a linear polarizer and a detector; fromthe Fourier coefficients of the time-varying signal, the incident Stokesvector can be determined [12, 30].

An experiment was carried out using the setup depicted in 4b includingboxed components (i) and (ii). A set of randomly selected linearpolarizer (LP) and quarterwave-plate angles (QWP) θ_(LP) and θ_(QWP) areselected. In an automated measurement, the mounts holding the LP and QWPmove to these pre-determined angles and the polarization state producedat each of these configurations is deduced using the meta-gratingpolarimeter. Next, the commercial rotating waveplate polarimeter (RWP)is placed in the beampath in place of the metasurface-gratingpolarimeter. The QWP and LP revisit the same positions and thepolarizations reported by the RWP are recorded.

The comparison is summarized in FIG. 5 with regards to the quantities ofazimuth and ellipticity of the polarization ellipse (plot is made of thedouble azimuth and ellipticity angles 2θ and 2∈, since these are theangular coordinates on the Poincaré sphere) and DOP. The graphs in thetop row of FIG. 5 plot the values reported by the metasurface-gratingpolarimeter along the vertical axis and the values reported by the RWPon the horizontal axis (if the two agreed exactly, all data points wouldlie along the black 1:1 correspondence line).

For each quantity, the difference in the values reported by the twopolarimeters is calculated and plotted in a histogram in the bottom rowof FIG. 5 . Each is fitted with a normal distribution, whose meandifferences (μ) and standard deviations (σ) are shown.

Discussion Parallel Polarization State Generation

As illustrated in FIG. 3 , for both the four polarization andtetrahedron gratings, the polarization ellipses observed experimentallyon the diffraction orders compellingly match both the desired targetellipses and those expected from FDTD simulation. A more quantitativeview of this comparison is provided herein, and in particular theaverage deviation in azimuth and ellipticity between target andmeasurement of 4.37% and 3.57%, respectively, was observed.Additionally, as shown herein, much of this difference may probably beattributed to fabrication imperfections and unpredictability of theexact element dimensions. As the nominal dimensions of the fabricatedgeometries are adjusted to test this effect, the measured polarizationellipses are observed to change smoothly (supplement). Given evenperfect fabrication, the ellipses would not completely match the targetstates since the optimization will never achieve perfection. From theperspective of polarimetry, at least, an imperfection in the performanceof the tetrahedron grating may be accounted for by the calibration(while the condition number of A may increase somewhat). The belowdisclosure discusses limits of the scheme and suggestions forimprovement.

Parallel Polarimetry

Above, a characterization of the metagrating's polarimeterfunctionality, specifically its ability to measure partially polarizedlight and its polarimetric performance in comparison with a commercialRWP has been presented. The calibration of the polarimeter may be asomewhat technical issue. The disclosure below discusses severalcalibration-related issues which could compromise the accuracy of thepolarimeter and suggest means for their improvement.

The meta-grating polarimeter can detect partially polarized lightproduced by a polarization Mach-Zehnder interferometer. The dependenceof DOP on linear polarizer angle follows the expected theoretical trend.At 45°, a minimum DOP of 1.2% with an uncertainty of 0.176% wasmeasured. While this ideally ought to be 0%, the DOP aggregates errorfrom all four Stokes components, and the minimum value achievable is ina sense a commentary on the accuracy of the polarimetric system as awhole. As discussed in the below disclosure, there may be severalsubtleties to the production of partially polarized light with apolarization Mach-Zehnder interferometer, including the DOP varying overthe profile of the output beam. The non-zero DOP observed at 45° couldbe a consequence of errors in the polarimeter, actual deviations of thebeam's DOP from zero, or more likely some combination of the two. Theresult, however, should be taken as a testament to the flexibility ofthe presented device—a single optical element in a completely parallelmeasurement can provide information about DOP, a coherence property ofthe light.

Lastly, the performance of the meta-grating polarimeter to that of acommercial rotating waveplate device was compared. For the quantities ofDOP, azimuth, and ellipticity—polarimetric parameters of commoninterest—the disclosure examined the difference in the values reportedby the two polarimeters and treated these as statistical quantities. ForDOP, the disclosure observed a standard deviation of σ=1.6% and a meandifference of μ=0.6%, a systematic error which could easily beattributed to one polarimeter being slightly misaligned. For azimuth χand ellipticity ϵ, σ=0.023 rad=1.32° and σ=0.0075 rad=0.43°,respectively, were observed. This may implicitly assume that the RWP isan absolute polarization reference; any degree to which this is not truewill increase the perceived error of the meta-grating polarimeter.Moreover, the error is itself polarization-dependent (see the belowdisclosure). By sampling the error more or less uniformly over allpossible polarizations, the values of σ for each parameter representworst-case performance. Already, these are in the vicinity of the errorsquoted for the RWP used (see the below disclosure). Notably, theperformance of the RWP may be matched with a device having no movingparts, no bulk birefringent polarization optics, and detector-limitedtime resolution.

The below disclosure reports the study of the effect ofangle-of-incidence on the polarimeter and concludes that up to about a±5° accidental misalignment, the polarimeter could still be used withreasonable accuracy.

Technological Perspective

In the present disclosure, polarimetry functionality has been embeddedin a single, flat metasurface where the two phase profiles may beapplied in the same plane, significantly improving prospects forwidescale application. Using just a linear polarizer (which may be, forexample, easily integrated on top of the device as a wire grid), asingle device may generate/measure polarization in parallel with no bulkbirefringent optics, permitting ease of integration. The device may beextended to spectroscopic polarimetry if linear arrays of detectors areused, or to polarization imaging if the detectors are replaced withimaging sensors. This may represent a far simpler solution to integratedfull-Stokes polarimeters and polarization cameras which would notinvolve bulk lithographic patterning of dichroic or birefringentmaterial on top of a focal plane array [5, 48, 49].

BRIEF SUMMARY OF DISCLOSURE SO FAR

The present disclosure provides for a method for designing a metasurfacediffraction grating with orders whose polarization states may bearbitrarily specified. Two such gratings were designed and fabricatedwith the polarization states of the diffraction orders beingcharacterized, finding close correspondence with desired targetpolarization states. The grating, by symmetry, may also function as aparallel analyzer of polarization, and may permit for snapshotfull-Stokes polarimetry. The grating's ability to measure partiallypolarized light was demonstrated. Additionally, the grating'sperformance was compared to a commercial RWP. A statistical analysisshows that the grating's accuracy is comparable to the one of thecommercial RWP. The metasurface-based polarimeter does not necessarilyinvolve either moving parts or bulk birefringent optics, which mayfacilitates its integrability, and thus may present a significantsimplification in polarimetric technology. The use of the grating may beextended to polarization imaging if detector arrays are used instead.Being a grating, the chromatic dispersion of the orders may also beharnessed for use in spectroscopic polarimetry.

Parallel Polarization State Production Optimization Procedure

A sketch of the optimization procedure is given in the presentdisclosure, which is resummarized here. Inputs include:

-   -   a set of {        } diffraction orders of length P whose polarization states are        to be controlled by the metasurface;    -   for each of the P orders, a Jones vector {right arrow over        (j)}^((m)) specifying the desired polarization on order m{        }. The Jones vector {right arrow over (j)}^((m)) is        parameterized by the quantities χ^((m)) and

${{\phi^{(m)}\mspace{14mu}{as}\mspace{14mu}{\overset{\rightarrow}{J}}^{(m)}} = \begin{bmatrix}{\cos\;\chi^{(m)}} \\{\sin\;\chi^{(m)}e^{i\;\phi^{(m)}}}\end{bmatrix}};$

-   -   the relative intensities of the beams on each diffraction order;    -   the Jones vector of the incident polarization, {right arrow over        (E)}₀; and    -   the number of phase-shifting elements Q to be included in the        grating.

Gradient Descent Optimization

The relative magnitudes and desired polarization states form a set ofconstraints on the Fourier coefficients of the ultimate phase-grating.

Both x- and y-polarized light experience independent phase gratings.These grating coefficients are a function of the phase profilesϕ_(x)({tilde over (x)} and ϕ_(y)({tilde over (x)} of the gratings, as afunction of the spatial coordinate {tilde over (x)}:

$\begin{matrix}{{{c_{x}^{(m)}\left( {\phi\left( \overset{\sim}{x} \right)} \right)} = {\left\langle m \middle| e^{i\;{\phi_{x}{(\overset{\sim}{x})}}} \right\rangle = {\frac{1}{2\pi}{\int_{0}^{d}{e^{i\;{\phi_{x}{(\overset{\sim}{x})}}}e^{i\frac{2\pi\; m\;\overset{\sim}{x}}{d}}d\;\overset{\sim}{x}}}}}}{and}} & (1) \\{{c_{y}^{(m)}\left( {\phi\left( \overset{\sim}{x} \right)} \right)} = {\left\langle m \middle| e^{i\;{\phi_{y}{(\overset{\sim}{x})}}} \right\rangle = {\frac{1}{2\pi}{\int_{0}^{d}{e^{i\;{\phi_{y}{(\overset{\sim}{x})}}}e^{i\frac{2\pi\; m\;\overset{\sim}{x}}{d}}d\;\overset{\sim}{x}}}}}} & (2)\end{matrix}$

The polarization and energy in each diffraction order is dictated by thefunctionals c_(x) and c_(y) ^(m)—the goal is to direct as much energy aspossible into the diffraction orders of interest, e.g. the set {

}. As such, optimization is made of the quantity

$\begin{matrix}{{\eta\left( {{\overset{\rightarrow}{\Phi}}_{x},{\overset{\rightarrow}{\Phi}}_{y}} \right)} = {\sum_{m\; \in {\{\ell\}}}\sqrt{\left( {c_{x}^{(m)}\left( {\overset{\rightarrow}{\Phi}}_{x} \right)} \right)^{2} + \left( {c_{y}^{(m)}\left( {\overset{\rightarrow}{\Phi}}_{y} \right)} \right)^{2}}}} & (3)\end{matrix}$under the constraints

$\begin{matrix}{{\frac{c_{y}^{(m)}}{c_{x}^{(m)}} = {\tan\;\chi^{(m)}}}{and}} & (4) \\{{\delta_{x}^{(m)} - \delta_{y}^{(m)}} = \phi^{(m)}} & (5)\end{matrix}$

This is best illustrated with an example from the present disclosure, agrating that produces −45° linear, right-hand circular, left-handcircular, and 45° linear polarizations on the m=−2, −1, +1, +2diffraction orders, respectively. The design of certain gratings in thepresent disclosure sought to distribute power equally among the fourorders of interest as well.

Certain designs in the present disclosure divided the gratingcoefficients c_(x) ^(m) and c_(y) ^(m) into real and imaginary parts asc_(x) ^((m))=|c_(x) ^((m))|e^(iδ) ^(x) ^((m)) and c_(y) ^((m))=|c_(y)^((m))|e^(iδ) ^(y) ^((m))

The constraints are, assuming that the incident light has a polarizationof

${\frac{1}{\sqrt{2}}\begin{bmatrix}1 \\1\end{bmatrix}}\text{:}$

c_(x)⁽⁻²⁾ = c_(y)⁽⁻²⁾ δ_(y)⁽⁻²⁾ − δ_(x)⁽⁻²⁾ = πc_(x)⁽⁻¹⁾ = c_(y)⁽⁻¹⁾${\delta_{y}^{({- 1})} - \delta_{x}^{({- 1})}} = \frac{\pi}{2}$c_(x)⁽⁺¹⁾ = c_(y)⁽⁺¹⁾${\delta_{y}^{({+ 1})} - \delta_{x}^{({+ 1})}} = {- \frac{\pi}{2}}$c_(x)⁽⁺²⁾ = c_(y)⁽⁺²⁾ δ_(y)⁽⁺²⁾ − δ_(x)⁽⁺²⁾ = 0 $\begin{matrix}{\sqrt{\left( c_{x}^{({- 2})} \right)^{2} + \left( c_{y}^{({- 2})} \right)^{2}} = \sqrt{\left( c_{x}^{({- 1})} \right)^{2} + \left( c_{y}^{({- 1})} \right)^{2}}} \\{= \sqrt{\left( c_{x}^{({+ 1})} \right)^{2} + \left( c_{y}^{({+ 1})} \right)^{2}}} \\{= \sqrt{\left( c_{x}^{({+ 2})} \right)^{2} + \left( c_{y}^{({+ 2})} \right)^{2}}}\end{matrix}$

Note that the final constraint asserts that optical power should beequally distributed among the different diffraction orders (since theincident polarization is linearly polarized at 45°, the weights on thec_(x) and c_(y) are equal).

The power in the orders of interest (Eq. 3) can then be optimized usinggradient descent, under these constraints using standard numericalpackages.

The results of this gradient descent optimization are given in FIGS. 6and 7 . In FIG. 6 , it can be seen that the theoretical efficiency ofthe phase grating η reaches a peak value of 73.1%.

Mathematical Analysis of Polarization Ellipses Produced by OptimizedPhase Grating

From the optimization procedure above, optimized x- and y-polarizationphase gratings producing the desired polarization states on thediffraction orders were obtained.

These phase profiles can be passed through a Fourier transform and theresultant polarization ellipses can be examined; the result is shown inFIG. 8 .

Unsurprisingly, the polarization ellipses derived mathematically fromthe optimized phase profiles match what is desired perfectly. This couldhave been gathered from FIG. 7 , since the constraints are matchednearly perfectly.

Material Implementation

With the optimized phase profiles ϕ_(x)({tilde over (x)}) andϕ_(y)({tilde over (x)}) so obtained, a structure implementing thecorrect phase on each of x- and y-polarized light at each gratingposition may be designed.

This has been discussed, for example, in [1 and 2, for this and otherreferences see section Literature below]. The approach is also brieflysummarized here.

A wide range of rectangular pillar structures of varying perpendiculartransverse dimensions w_(x) and w_(y) are simulated assuming plane waveillumination and periodic boundary conditions using the FiniteDifference Time Domain method (FDTD). Using far-field projection, thetransmission amplitude for each polarization (t_(x) and t_(y)) and theeigen-phase shifts on each polarization (ϕ_(x) and ϕ_(x)) aredetermined. That is, knowledge of w_(x) and w_(y) specifies thephase-shifting and transmission properties of the element throughsimulation.

Here the interest is in the opposite problem wherein one wishes to knowthe w_(x)/w_(y) best implementing a specified ϕ_(x) and ϕ_(y). The bestelement can be located by finding the element with the amplitudetransmission (as close as possible to 1) while minimizing the deviationof the implemented phase from the desired one. For a more completedescription of this, see the supplement to [50].

This allows one to straightforwardly transform an optimized set ofgrating phase profiles {right arrow over (Φ)}_(x) and {right arrow over(Φ)}_(y) into a real geometry to be fabricated in a material of interest(in the present case, TiO₂).

FDTD Simulation of Designed Gratings

The periodicity defining a grating may be fortunate from a simulationpoint of view, and the optimized gratings comprising of 20 elementsnaturally lend themselves to simulation. Thus, the optimized gratingsmay be directly simulated and the polarization ellipses examined.

This was performed using plane wave illumination and periodic boundaryconditions, using the geometry shown in FIG. 9 .

In FIG. 10 , polarization ellipses expected from simulation are shownfor the four-polarization grating.

Discrepancy Between Simulation And Target

As stated above, the phase profiles φ_(x)({tilde over (x)}) andφ_(y)({tilde over (x)}) are optimized and found under the assumptionthat at each point along the grating, amplitude transmission is uniform.In the selection of elements to implement these phase profiles, a bestattempt is made to find an element implementing the requisite φ_(x) andφ_(y) at each position with amplitude transmission as close as possibleto unity.

Using FDTD simulation, a library of phase-shifting structures can beassembled in which phase-shifting geometries (e.g., w_(x) and w_(y)) areassociated with phase shifts and amplitude transmissions. The extent towhich it is possible to realize any desired set of phases {φ_(x), φ_(y)}with nearly unity transmission depends on the diversity of structuresavailable in a desired material implementation. Once desired phaseprofiles are converted into such a “best fit” grating geometry, thephase shifts and amplitude transmissions of the structured chosen fromthe library can be assessed.

This is shown in FIG. 33 . It can be seen that at some points along thex- and y-polarization phase gratings, the transmission dips as low as70%, while at other positions it is indeed near unity. The assumption bythe optimization algorithm of unity transmission, then, is broken at thetime its results are converted into a real grating design.

Certain non-uniformity in amplitude transmission can easily beincorporated into the Fourier transform and computation of the meritfunction Eq. 3.

Every set of phase profiles during at any step during the optimizationprocess has an associated “best fit” geometry. Our original solution tothis issue, then, was to include an examination of the transmission andphase profiles of this “best fit” grating geometry as part of theoptimization process and merit-figure computation. The optimization,however, would often not converge. Instead, a purely mathematicalprocedure based on gradient descent is used to obtain φ_(x)({tilde over(x)}) and φ_(y)({tilde over (x)}), regardless of any particular materialimplementation. Then, this result is used as the initial condition for agradient-free scheme (namely, pattern search) that improves this resulttaking limitations of our library of structures explicitly into accountby computing the “best fit” grating geometry at each step of theoptimization process.

Generally, the more diffraction orders one wishes to control, the lesssuccessful the optimization will be, and there are certain situations inwhich the optimization could become over-constrained.

In some embodiments of this disclosure, focus is made on gratings inwhich four diffraction orders possess tailored and controlledpolarization states. This is because the Stokes vector itself possessesfour elements, so analysis on four of these diffraction orders issufficient for the construction of a full-Stokes polarimeter. However,as an illustration of the limits of the optimization scheme, it shouldbe noted that a grating controlling six diffraction orders was designed.In particular, it is sought to create a metasurface polarization gratingin which the inner six diffraction orders would contain the cardinaldirections on the Poincare sphere—that is, RCP, LCP, 45°, 135°, x, andy.

The results are shown in FIG. 34 . First, optimized phase profilesφ_(x)({tilde over (x)}) and φ_(y)({tilde over (x)}) are found usinggradient descent alone, without reference to any properties of actualphase-shifting structures. Mathematically, then, it is possible to findphase profiles producing the desired polarization ellipses for thiscase.

However, when the optimization is carried out in two steps (as describedabove, using pattern search and actual phase-shifter properties) and theresultant grating is simulated, the ellipses are not generally producedwithout significant distortion. This is shown in the bottom of FIG. 34 .By expanding the library of available structures and improvement of theoptimization scheme, it may be possible to expand to the control of manymore diffraction orders.

Fabrication of Structures

The gratings themselves are fabricated using a titanium dioxide processdeveloped for visible frequency metasurfaces. It is very brieflysummarized here.

A positive electron beam resist is spun on a glass substrate and thegrating pattern is exposed using electron beam lithography. The resistis developed and an amorphous TiO₂ film is deposited in the voids usingatomic layer deposition (ALD). Excess TiO₂ is etched back using reactiveion etching, and finally the electron beam resist is removed using asolvent, yielding free-standing TiO₂ pillars.

The conformality of the ALD process as well as the use of the resist asa template for the pattern yields high aspect ratio structures withnear-vertical sidewalls.

Measurement of Polarization Ellipses

The fabricated gratings are mounted on a translation mount andilluminated with a beam of green laser light linearly polarized at 45°with respect to the Cartesian axes of the grating and pillar elements.The light disperses into a multitude of grating orders, each of whichcan be characterized with a commercial polarimeter (the same polarimeterthat is used for comparison in the disclosure).

It is particularly important that the incident light is polarized at 45°relative to the x/y coordinate system of the grating, because the entiredesign procedure is predicated on the incident light having thispolarization. A misalignment of this linear polarization, then, wouldyield undue experimental error. A two step procedure is undertaken tomutually align the grating and incident polarization:

Alignment of the linear polarizer to the optical table: A Glan-Thompsonlinear polarizer was used to linearly polarize the laser light. Theplane of the optical table was used as a reference coordinate system forthe experiment. Using the same commercial rotating waveplate polarimeteremployed in the experiments, which presumably is itself well-calibratedwith respect to the table on which it is mounted, the linear polarizerwas turned until the polarimeter revealed that the light is polarized at45°.

Alignment of the grating to the optical table: The axes of the gratingshould themselves be aligned to the plane of the optical table. Theplane of the grating orders which fan out in space provide a referenceto the orientation of the metasurface grating. The grating is placed ina freely rotating mount and an iris is used to assure that all gratingorders are at a consistent height above the table.

Effect of Fabrication Imperfections on Produced Polarization Ellipses

Electron beam lithography comes with its own inherent complicationswhich cause size discrepancies between desired structures (as in a CADlayout file) and what is actually realized. Generally, the fabricatedstructures are larger than intended. The phase imparted by thestructures in this disclosure is size-dependent, so one would expectfabrication imperfections to have a notable effect on the polarizationellipses observed on the diffraction orders, relative to design.

In view of this effect, for each grating design, several samples werefabricated with either fixed size offsets in the CAD (10, 20, or 30 nmsmaller than desired) or fixed size scaling factors (all dimensionsscaled by 85, 75, or 70%). The polarization states on the diffractionorders from all such gratings were recorded.

The polarization ellipses recorded from the gratings with different sizeoffsets and scalings are shown in FIGS. 11 and 12 , for thefour-polarization and tetrahedron gratings, respectively.

It can be seen that varying the nominal size of the fabricatedstructures has a significant effect on the observed polarizationellipses, which is to be expected. The data presented in the text camefrom samples with −20 nm offset for the four-polarization grating and−30 nm size offset for the tetrahedron grating; these samples producedellipses closest to the target polarizations.

Extended Results Power on Diffraction Orders

Tabulated Data on Measured Polarization Ellipses

Tetrahedron Grating % Quantity Designed FDTD Measured Difference m = −2order Azimuth angle θ 0 −0.0092 −0.0995 3.17 Ellipticity angle ϵ 0.16990.2049 0.3066 2.23 m = −1 order Azimuth angle θ 1.0472 1.0399 0.87895.36 Ellipticity angle ϵ 0.1699 0.2168 0.3191 9.50 m = +1 order Azimuthangle θ −1.0472 −1.2464 −1.3687 10.23 Ellipticity angle ϵ 0.1699 0.10890.2165 2.97 m = +2 order Azimuth angle θ undefined 0.3589 −0.0218 N/AEllipticity angle ϵ −0.7854 −0.7568 −0.7244 3.88 Four PolarizationGrating % Quantity Designed FDTD Measured Difference m = −2 orderAzimuth angle θ −0.7854 −0.7867 −0.8840 3.14 Ellipticity angle ϵ 0−0.0026 −0.0299 1.90 m = −1 order Azimuth angle θ undefined −0.26381.4019 N/A Ellipticity angle ϵ 0.7854 0.7212 0.7462 2.50 m = +1 orderAzimuth angle θ undefined 1.4905 −0.0428 N/A Ellipticity angle ϵ −0.7854−0.7432 −0.7249 3.85 m = +2 order Azimuth angle θ 0.7854 0.7912 0.72821.82 Ellipticity angle ϵ 0 −0.0297 −0.0274 1.74

Metasurface Polarimetry

This section discusses the use of a metasurface polarization grating asa full-Stokes polarimeter, that is, as a sensor allowing the detectionof incident polarization state based on the intensities measured on eachof the four channels. These channels are the beams diffracted into them=−2, −1, +1, and +2 diffraction orders, after passing through a linearpolarizer oriented at 45°.

Details of Optical Setup

Of particular importance in polarimetry is the quality of polarizationoptics used in calibration. After all, uncertainty about thepolarizations used for calibration degrades the ultimate accuracy of anycalibrated polarimeter. Of course, absolute polarization references arefew-and-far-between [53]. Perhaps the absolute polarization statereferences that exist are those of atomic transitions, constrained byquantum mechanical selection rules to take on a certain polarizationstate. Thus, in the field of polarimetry, polarimeters are oftenassessed by comparison with pre-established polarimeters.

Nevertheless, some certainty about the polarization optics used wasestablished.

Polarizers

Film-based polarizers may be the least expensive options and lendthemselves to mass production. For example, for a commercially availablefilm polarizer from the vendor ThorLabs (part no. LPVISE100-A), the partspecifications claim an intensity extinction ratio of approximately8000. Given this, it was initially throught that these would besufficient for this project's specifications. However, upon testing, itwas found that the intensity extinction ratio of these film polarizersis often well below 1000, sometimes on the order of hundreds. Forpolarimetry, especially the presently considered metasurface-basedpolarimeter, this may be a significant problem. The fundamental quantityis the electric field, which goes as the square root of the intensity.If the intensity extinction ratio is on the order of hundreds, theelectric field extinction ratio is on the order of tens. This may beunacceptable.

Whereas film polarizers rely on inherent material dichroism, a secondpolarizer technology relies on birefringent crystalline materials. Forexample, Glan-Thompson (and Glan-Taylor) polarizers sandwich twobirefringent crystals together in a beamsplitter-like configuration. Thetwo are cut so that one linear polarization transmits through relativelyunattenuated, while the other totally internally reflects. Since thepolarizing mechanism here uses total internal reflection rather thansome property subject to an artificial medium, the extinction ratios maybe much higher. For instance, the primary polarizer used in all thepolarimetry and polarization state measurements (ThorLabs part no.GTH5M-A) has a specified intensity extinction ratio of 100000. While theextinction ratio of the present unit was below this, it performedsignificantly better than a comparable film polarizer.

In any measurement in the text in which a linear polarizer is rotated,as well as in all the measurements testing the polarization-stategeneration capabilities of the metasurface, the above-mentionedGlan-Thompson polarizer was used.

Quarter-Wave Plates

In optics texts, quarter-(and half-)wave plates are usually presented asmathematical objects, possessing the special property of exactly

$\frac{\pi}{2}$(or π) retardance. A subtlety often overlooked by many practitioners ofoptics is that this is seldom the case. Waveplate manufacturing involvesthe grinding of bulk birefringent crystal with wavelength-scaleprecision.

Broadly, waveplates may be classified as either zero-order ormulti-order. When the crystal is manufactured in a waveplate cut [54],one of the crystalline axes is aligned with the intended propagationaxis of the waveplate. Then, in plane, orthogonal axes experience arefractive index difference of Δn due to the crystal's birefringence. Ina zero-order waveplate, the thickness of the crystal is given by

${t = \frac{\delta\;\lambda}{2\pi\;\Delta\; n}},$where δ is the desired retardance and λ is the design wavelength. Thatis, the thickness may be the minimum that still imparts the correctphase retardation on the wavefront. In a multi-order waveplate, on theother hand, the thickness of the crystal is given by

$t = \frac{\left( {{2\pi\; N} + \delta} \right)\lambda}{2\pi\;\Delta\; n}$where N is the “order” of the waveplate.

Higher order waveplates tend to be less accurate, and of course have ahigher dispersion of the retardance δ with wavelength. Some of theinaccuracy in zero-order waveplates can be appreciated by examining themeans by which they are fabricated—in many cases, the grinding of acrystal waveplate should be stopped and started periodically so that theretardance can be monitored by a polarimetric instrument [54].

Consequently, multi-order quarter-wave plates commonly deviate by tendegrees or more from the desired retardance. This is exacerbated forwaveplates that attempt to operate over a broadband. Even zero-orderwaveplates deviate by up to 3°-4° from the desired retardance, in ourexperience.

Precise knowledge of the properties of the polarization optic being usedmay be desired in particular during calibration. In the present case, atechnique allowing for imperfections of the quarter-waveplate wasemployed, so long as the deviation from 90° retardance is small [55].Zero-order waveplates designed for the wavelength of interest (λ=532 nm)fit this criterion, and as such, a zero-order waveplate from ThorLabs(part no. WPQ10M-532) was used. Any measurement that involves turning aquarter-waveplate, in particular the calibration and comparison to theRWP in the present disclosure, are made using this waveplate.

Other Polarization Optics

The most crucial polarization optics may be the main linear polarizerand quarter-waveplate used during calibration.

Several other polarization optics were also involved in present setup.The first is a quarter-waveplate immediately after the laser output. Thelaser source has a preferential linear polarization, so a waveplate wasinserted in order to (roughly) circularize the source polarization. Thisway, as the linear polarizer is varied, power variations are minimized.As this circularization need not be exact by any means, most anyquarter-waveplate can be used. ThorLabs part no. AQWPO5M-600 was readilyon hand, an achromatic waveplate in the visible region, whose retardanceis estimated by crude means to be approximately 75° at λ=532 nm.

Finally, a polarizer was used behind the metasurface grating. Thispolarizer was oriented at 45° relative to the metasurface grating(though, in reality, this angle could be wrong and its effect of apotential error can be calibrated away). The integrity of this polarizeris not nearly as important as that of the polarizer used forcalibration. In some embodiments, it may be necessary to have apolarizer-like element, as discussed below, for the functioning of thegrating as a polarimeter. However, if an extinction ratio for suchpolarizer-like element is not so high, it may still be used because ifthe same polarizer is placed behind the metasurface during bothcalibration and the actual measurement, the polarizer's effect may becompensated.

Therefore, a sheet polarizer, ThorLabs part no. LPVISE100-A was used forthat purpose. The grating orders diverge in angle sufficiently fast thatusing a thick Glan-Thompson polarizer without first collimating thediffraction orders using a lens would present a technical challenge.Therefore, the film polarizer is ideal because all four orders may passthrough its 1″ clear aperture. Of course, each passes through at anon-normal angle, but since these angles are constant, that effect, too,may be absorbed into the calibration.

Detection Electronics

It may be preferred to use a detector or sensor, which is linear inintensity at a design wavelength, such as λ=532 nm in the actualexperiments. Moreover, if no fast polarization modulations are involved,it may be sufficient to monitor the intensity in DC. For example, in thepresent disclosure, a silicon photodiode, with an optional amplificationscheme could be sufficient. In some embodiments, it may be preferred touse a detector with a larger detection area, such as greater than 1 mm²or greater than 2 mm² or greater than 5 mm² or greater than 7 mm² orgreater than 10 mm². The use of a detector with a larger detection areamay, for example, simply the alignment of optical elements.

In particular, this disclosure used Hamamatsu part no. S1223-01, asilicon photodiode with an active area of 3.6 mm×3.6 mm—the large areaeases alignment constraints.

Photocurrent from each of the four photodiodes was amplified using astandard transimpedance amplifier. The amplified value is read by a14-bit analog-to-digital converter (ADC) from National Instruments,transmitted serially over USB, and then recorded on a PC. Given the slowtime scale of the measurements (DC, essentially) no particular detail ofthis configuration is crucial.

Beam Re-Sizing Optics

Though not depicted or mentioned in the present disclosure, twoconventional lenses, with focal lengths of roughly 50 and 25 mm, toshrink the beam-waist before illuminating the metasurface. Of course,with electron-beam lithography, there are constraints on the size of ametasurface that can ultimately be fabricated. Since the grating usedfor polarimetry is 1.5 mm×1.5 mm, the beam was shrunk so that its entireextent may reside inside of the fabricated metasurface. Preferably, evenif the beam moves around as polarization optics are rotated, it willremain inside of the metasurface.

The beam re-sizing lenses were deliberately placed before thepolarization optics that modify the polarization of the beam. Lenses mayhave some inherent, polarization-modifying stress-birefringence. Placingthem before the final polarization optics may allow neglecting this, ifit does exist.

The Polarizer Following the Grating

The present disclosure presents a design scheme for metasurfacepolarization gratings, that, when illuminated with a prescribedpolarization state may produce desired states of polarization on itsdiffraction orders. By the reciprocity of polarization generators andanalyzers, when the grating is used in reverse (that is, light passesthrough the grating and then a polarizer), the grating may also be usedas a polarimeter.

It might seem that the grating itself, without the inclusion of thelinear polarizer, could function as a polarimeter alone. After all, asdetailed in the present disclosure, each grating order can be thought ofas a diattenuator and a waveplate in series. These alone, however, maynot be sufficient to form a full-Stokes polarimeter without the linearpolarizer.

As detailed in the text, diffraction order m may be thought of as havingits own characteristic Jones matrix given by

$\begin{matrix}{J_{m} = {\begin{pmatrix}c_{x}^{m} & 0 \\0 & c_{y}^{m}\end{pmatrix}\begin{pmatrix}e^{i\;\delta_{x}^{m}} & 0 \\0 & e^{i\;\delta_{y}^{m}}\end{pmatrix}}} & (6)\end{matrix}$

For a more general view of the behavior of the diffraction order,including its response to partially and un-polarized light, attentionturns to the Mueller calculus, that is, the theory of 4×4 matrixoperators on Stokes (rather than Jones) vectors [5, 7].

The Mueller matrix of a diattenuator oriented at 0° with respect to thex/y coordinate system is given by

$\begin{matrix}{\frac{1}{2}\begin{pmatrix}{c_{x}^{m} + c_{y}^{m}} & {c_{x}^{m} - c_{y}^{m}} & 0 & 0 \\{c_{x}^{m} - c_{y}^{m}} & {c_{x}^{m} + c_{y}^{m}} & 0 & 0 \\0 & 0 & {2\sqrt{c_{x}^{m}c_{y}^{m}}} & 0 \\0 & 0 & 0 & {2\sqrt{c_{x}^{m}c_{y}^{m}}}\end{pmatrix}} & (7)\end{matrix}$

where c_(x) ^(m) and c_(y) ^(m) are the amplitude transmissioncoefficients along x and y. The Mueller matrix of a waveplate alsooriented in this way is given by:

$\begin{matrix}\begin{pmatrix}1 & 0 & 0 & 0 \\0 & 1 & 0 & 0 \\0 & 0 & {\cos\;\delta_{m}} & {\sin\;\delta_{m}} \\0 & 0 & {{- \sin}\;\delta_{m}} & {\cos\;\delta_{m}}\end{pmatrix} & (8)\end{matrix}$

Here, δ_(m)=δ_(x) ^(m)−δ_(y) ^(m) is the retardance of the waveplate.

The composite Mueller matrix of diffraction order m, then, is given byM_(m) the product of the two above matrices:

$\begin{matrix}{M_{m} = {\frac{1}{2}\begin{pmatrix}{c_{x}^{m} + c_{y}^{m}} & {c_{x}^{m} - c_{y}^{m}} & 0 & 0 \\{c_{x}^{m} - c_{y}^{m}} & {c_{x}^{m} + c_{y}^{m}} & 0 & 0 \\0 & 0 & {2\cos\;\delta_{m}\sqrt{c_{x}^{m}c_{y}^{m}}} & {2\;\sin\;\delta_{m}\sqrt{c_{x}^{m}c_{y}^{m}}} \\0 & 0 & {{- 2}\;\sin\;\delta_{m}\sqrt{c_{x}^{m}c_{y}^{m}}} & {2\;\cos\;\delta_{m}\sqrt{c_{x}^{m}c_{y}^{m}}}\end{pmatrix}}} & (9)\end{matrix}$

M_(m) gives the full-polarization sensitive behavior of the combinationof a diattenuator and waveplate on diffraction order m. What is truly ofinterest is the first row—it is this row that dictates, for a givenincident Stokes vector, what power will be measured at the output of theanalyzer. If four such analyzers are placed on four diffraction orders,say m=−2, −1, +1, and +2, akin to the polarimeter compared here, one canimmediately write the instrument matrix:

$\begin{matrix}{A = {\frac{1}{2}\begin{pmatrix}{c_{x}^{- 2} + c_{y}^{- 2}} & {c_{x}^{- 2} - c_{y}^{- 2}} & 0 & 0 \\{c_{x}^{- 1} + c_{y}^{- 1}} & {c_{x}^{- 1} - c_{y}^{- 1}} & 0 & 0 \\{c_{x}^{+ 1} + c_{y}^{+ 1}} & {c_{x}^{+ 1} - c_{y}^{+ 1}} & 0 & 0 \\{c_{x}^{+ 2} + c_{y}^{+ 2}} & {c_{x}^{+ 2} - c_{y}^{+ 2}} & 0 & 0\end{pmatrix}}} & (10)\end{matrix}$

By inspection, it is evident that the row-space of A does not span

⁴ and one can immediately conclude that det A=0. Inversion isimpossible, and this configuration cannot act as a full-Stokespolarimeter.

Let us consider a situation instead in which a polarizer is placed afterthe diattenuator/waveplate configuration. The polarizer is allowed totake on a general orientation angle relative to x/y which we denote θ.Its Mueller matrix is:

$\begin{matrix}{\frac{1}{2}\begin{pmatrix}1 & {\cos\; 2\;\theta} & {\sin\; 2\;\theta} & 0 \\{\cos\; 2\;\theta} & {\cos^{2}2\theta} & {\sin\; 2\theta\;\cos\; 2\;\theta} & 0 \\{\sin\; 2\;\theta} & {\sin\; 2\;\theta\;\cos\; 2\;\theta} & {\sin^{2}2\;\theta} & 0 \\0 & 0 & 0 & 2\end{pmatrix}} & (11)\end{matrix}$

If pre-multiplication is performed for M_(m) from above with the Muellermatrix of a polarizer oriented at θ, one obtains the Mueller matrix:

$\begin{matrix}{M_{m}^{\prime} = {\frac{1}{4}\begin{pmatrix}{\begin{matrix}{c_{x}^{m} + c_{y}^{m} +} \\\left( {c_{x}^{m} - c_{y}^{m}} \right)\end{matrix}\cos\; 2\;\theta} & {\begin{matrix}{c_{x}^{m} - c_{y}^{m} +} \\\left( {c_{x}^{m} + c_{y}^{m}} \right)\end{matrix}\cos\; 2\;\theta} & {2\sqrt{c_{x}^{m}c_{y}^{m}}\cos\;\delta_{m}\sin\; 2\theta} & {2\sqrt{c_{x}^{m}c_{y}^{m}}\sin\;\delta_{m}\sin\; 2\theta} \\{\cos\; 2\;{\theta\begin{pmatrix}{c_{x}^{m} + c_{y}^{m} +} \\\left( {c_{x}^{m} - c_{y}^{m}} \right)\end{pmatrix}}\cos\; 2\;\theta} & {\cos\; 2\;{\theta\begin{pmatrix}{c_{x}^{m} - c_{y}^{m} +} \\\left( {c_{x}^{m} - c_{y}^{m}} \right)\end{pmatrix}}\cos\; 2\;\theta} & {\sqrt{c_{x}^{m}c_{y}^{m}}\cos\;\delta_{m}\sin\; 4\theta} & {\sqrt{c_{x}^{m}c_{y}^{m}}\sin\;\delta_{m}\sin\; 4\theta} \\{\left( {\begin{matrix}{c_{x}^{m} + c_{y}^{m} +} \\\left( {c_{x}^{m} - c_{y}^{m}} \right)\end{matrix}\cos\; 2\;\theta} \right)\sin\; 2\;\theta} & {\left( {\begin{matrix}{c_{x}^{m} - c_{y}^{m} +} \\\left( {c_{x}^{m} + c_{y}^{m}} \right)\end{matrix}\cos\; 2\;\theta} \right)\sin\; 2\;\theta} & {2\sqrt{c_{x}^{m}c_{y}^{m}}\cos\;\delta_{m}\sin^{2}\; 2\theta} & {2\sqrt{c_{x}^{m}c_{y}^{m}}\sin\;\delta_{m}{\sin\;}^{2}2\theta} \\0 & 0 & {{- 4}\sqrt{c_{x}^{m}c_{y}^{m}}\sin\; 2\theta} & {4\sqrt{c_{x}^{m}c_{y}^{m}}\cos\;\delta_{m}}\end{pmatrix}}} & (12)\end{matrix}$

Notably, the first row of this matrix has components in all four of itsentries that are, in general, non-zero. An instrument matrix from foursuch analyzers, all sharing the same polarizer at the same orientationθ:

$A^{\prime} = \begin{pmatrix}{c_{x}^{- 2} + c_{y}^{- 2} + {\left( {c_{x}^{- 2} - c_{y}^{- 2}} \right)\cos\mspace{14mu} 2\theta}} & {c_{x}^{- 2} - c_{y}^{- 2} + {\left( {c_{x}^{- 2} + c_{y}^{- 2}} \right)\cos\mspace{14mu} 2\theta}} & {2\sqrt{c_{x}^{- 2}c_{y}^{- 2}}\cos\mspace{14mu}\delta_{- 2}\mspace{14mu}\sin\mspace{14mu} 2\theta} & {2\sqrt{c_{x}^{- 2}c_{y}^{- 2}}\sin\mspace{14mu}\delta_{- 2}\mspace{14mu}\sin\mspace{14mu} 2\theta} \\{c_{x}^{- 1} + c_{y}^{- 1} + {\left( {c_{x}^{- 1} - c_{y}^{- 1}} \right)\cos\mspace{14mu} 2\theta}} & {c_{x}^{- 1} - c_{y}^{- 1} + {\left( {c_{x}^{- 1} + c_{y}^{- 1}} \right)\cos\mspace{14mu} 2\theta}} & {2\sqrt{c_{x}^{- 1}c_{y}^{- 1}}\cos\mspace{14mu}\delta_{- 1}\mspace{14mu}\sin\mspace{14mu} 2\theta} & {2\sqrt{c_{x}^{- 1}c_{y}^{- 1}}\sin\mspace{14mu}\delta_{- 1}\mspace{14mu}\sin\mspace{14mu} 2\theta} \\{c_{x}^{+ 1} + c_{y}^{+ 1} + {\left( {c_{x}^{+ 1} - c_{y}^{+ 1}} \right)\cos\mspace{14mu} 2\theta}} & {c_{x}^{+ 1} - c_{y}^{+ 1} + {\left( {c_{x}^{+ 1} + c_{y}^{+ 1}} \right)\cos\mspace{14mu} 2\theta}} & {2\sqrt{c_{x}^{+ 1}c_{y}^{+ 1}}\cos\mspace{14mu}\delta_{+ 1}\mspace{14mu}\sin\mspace{14mu} 2\theta} & {2\sqrt{c_{x}^{+ 1}c_{y}^{+ 1}}\sin\mspace{14mu}\delta_{+ 1}\mspace{14mu}\sin\mspace{14mu} 2\theta} \\{c_{x}^{+ 2} + c_{y}^{+ 2} + {\left( {c_{x}^{+ 2} - c_{y}^{+ 2}} \right)\cos\mspace{14mu} 2\theta}} & {c_{x}^{+ 2} - c_{y}^{+ 2} + {\left( {c_{x}^{+ 2} + c_{y}^{+ 2}} \right)\cos\mspace{14mu} 2\theta}} & {2\sqrt{c_{x}^{+ 2}c_{y}^{+ 2}}\cos\mspace{14mu}\delta_{+ 2}\mspace{14mu}\sin\mspace{14mu} 2\theta} & {2\sqrt{c_{x}^{+ 2}c_{y}^{+ 2}}\sin\mspace{14mu}\delta_{+ 2}\mspace{14mu}\sin\mspace{14mu} 2\theta}\end{pmatrix}$

There are suitable choices of the {c_(x) ^(m)}, {c_(y) ^(m)}, and{δ_(m)} such that A′ is invertible.

Calibration Procedure, Step-By-Step

This section details the calibration procedure used to calibrate themetasurface polarimeter in a very deliberate manner. The process usedhere may be viewed as a modification of that presented in [55].

Linear Polarizer

Following [55], one may write the instrument matrix to-be-determined interms of its columns:

$\begin{matrix}{A = \begin{pmatrix}| & | & | & | \\{\overset{\rightarrow}{A}}_{0} & {\overset{\rightarrow}{A}}_{1} & {\overset{\rightarrow}{A}}_{2} & {\overset{\rightarrow}{A}}_{3} \\| & | & | & |\end{pmatrix}} & (14)\end{matrix}$

In this first calibration step, the polarization incident on themeta-grating polarimeter is set by a single linear polarizer(Glan-Thompson). When the linear polarizer is oriented at an angle θ, itproduces a Stokes vector given by {right arrow over (S)}(θ)=[1 cos 2θsin 2θ 0]^(T) assuming that the power of the incident beam is unity.Then, the intensity vector (that is, the list of intensities measured onthe four photodiodes) is given by A{right arrow over (S)}(θ):{right arrow over (I)}={right arrow over (A)} ₀ +{right arrow over (A)}₁ cos 2θ+{right arrow over (A)} ₂ sin 2θ  (15)

As the incident linear polarization is varied, the values on each of thefour channels can be recorded. We can normalize these values by theincident beam power i_(θ) and plot against θ.

This yields four sinusoidal curves. By fitting the curves to thefunctional form of Eqn. 15, the first three columns of the instrumentmatrix are obtained (that is, {right arrow over (A)}₀, {right arrow over(A)}₁, and {right arrow over (A)}₂).

In FIG. 18 , the optical setup during this first stage of calibration isshown. Light from the laser source at λ=532 nm bounces off two alignmentmirrors and passes through a quarter-waveplate (as discussed above) tocompensate for the preferred linear polarization of the laser source. Itpasses through two beam reduction lenses so that its spatial extent mayfit inside the square boundaries of the 1.5 mm×1.5 mm metasurfacepolarization grating. After being polarized by a Glan-Thompson linearpolarizer, light passes through the metasurface diffraction grating andis split into many diffraction orders, the central four of which passthrough a fixed linear polarizer at 45° and then impinge upon fourphotodiodes, whose values are amplified and recorded on a computer.

The linear polarizer is placed in a motorized rotation mount and itsorientation is varied in 5° steps. At each step, a photodiode swings infront of the beam (blocking it), records the beam power, and then swingsback out in an automated fashion in order to obtain the incident beampower i_(θ). Additionally, for every θ, measurement is made of the powerreported by the four photodiodes directly following the metasurface.

It should be noted that this first calibration step introduces acoordinate system to the polarimeter—the θ=0° point of the linearpolarizer becomes the origin of the polarimeter's angular coordinatesystem. This need not be necessarily well-aligned with the optical tableor any external coordinate frame.

It was found that, as the linear polarizer rotated, its trajectory onthe metasurface sample moved around in a circular orbit, of sorts (as inFIG. 19 ). At times, the incident beam would near the edge of thesample. This “orbiting” effect compromised the measurements and wouldseverely skew the sinusoidal calibration curves. In order to allow afreedom to correct this effect, the Glan-Thompson polarizer was mountedin a tip/tilt mount before placing it in a motorized rotation mount.Then, the rotation mount was set to rotate at a constant angularvelocity and observe the aforementioned “orbit” effect and correct thetip/tilt of the polarizer until it was eliminated.

The source of this effect was not reliably identified. It may bepossible that the Glan-Thompson, being a prism polarizer, may sufferfrom some misalignment of the two prisms relative to one another.Consequently, there would be a refraction at the middle interface.

Linear Polarizer and Quarter-Waveplate

With just a linear polarizer, the first three columns of the instrumentmatrix are easily determined. The last column involves test polarizationstates with some chirality.

Circularly polarized light has a Stokes vector given by [1 0 0 ±1]^(T)where the ±distinguishes between right- and left-handed circularpolarization. As shown in [55], the last column of the instrument matrixmay be written as

$\begin{matrix}{{\overset{\rightarrow}{A}}_{3} = {\frac{1}{2}\left( {{\overset{\rightarrow}{I}}_{RCP} - {\overset{\rightarrow}{I}}_{LCP}} \right)}} & (16)\end{matrix}$

The (incident intensity normalized) values of the readings on all fourphotodiodes are to be obtained when exposed to each circularpolarization. This presents a problem however; as detailed above,production of perfect circular polarization—at least with off-the-shelfwaveplates—is nearly impossible.

Azzam presents a solution [55], so long as the retardance of thewaveplate is nearly a quarter-wave, that is, so long as it deviates from90° by a small angle. A linear polarization and a quarter-waveplateoriented at 45° relative to one another produce nearly circularpolarization. If both the polarizer and the waveplate are rotatedtogether by 90°, the same nearly-circular elliptical polarization willbe produced, but rotated by 90°. To first order, the effect of averagingthe polarimeter's readings when exposed to both configurations is thesame as if perfect circular polarization were available.

In Azzam's original scheme, this average contains two data points, at 0°and 90°. In the present disclosure, many such data points between 0° and360° may be taken in order to decrease error.

In the experiment, as depicted in FIG. 15 , both a linear polarizer anda quarter-waveplate are placed in the beampath in front of themetasurface grating polarimeter. Both are controlled with automatedrotation mounts. When the quarter-waveplate is inserted, the angularposition of its fast axis should be determined. A second Glan-Thompsonpolarizer is placed in front of the first, and the first polarizer isrotated until the intensity of the beam is nulled (the polarizers arecrossed). Then, the quarter-waveplate is inserted between the two, andangular position at which the intensity transmitted through thepolarizer-waveplate-polarizer configuration is maximized. This should bewhere the fast axis is at 45° relative to the first polarizer. Thisposition is determined by testing the transmitted intensity at manyangular locations and fitting a trigonometric function to find thelocation of the maximum.

Next, the second polarizer is removed, so that the first polarizer andthe waveplate oriented at a relative angle of 45° remain. Thepolarizer/waveplate combination is rotated in tandem in 5° incrementsand at each configuration, the intensities on the four photodiodes arerecorded. In addition, the power of the incident beam i_(θ) is alsorecorded in each configuration by a photodiode that moves in and out ofthe beam.

For each of the four photodiodes, the normalized intensities measured ateach θ are computed. These are averaged together to form the vector{right arrow over (I)}_(RCP). Then, the quarter-waveplate is rotated by90°, and the entire process above is repeated to obtain the vector{right arrow over (I)}_(LCP).

This data is depicted in FIG. 16 . It is of note that for incident RCPlight, the intensity on PD #4 is very low, while for LCP the case isreversed. This confirms that, indeed, one of the diffraction ordersproduces near-circularly polarized light, as desired. Second, it isnoted that the values on each photodiode vary as the linearpolarizer/QWP combination is rotated. This is to be expected to acertain extent, since the polarization is not exactly circular. Onewould expect, however, that the variation would be sinusoidal in nature.Despite our best efforts, we cannot account for the skewed shape ofthese curves; it may have something to do with the orbiting effect ofFIG. 14 .

Compilation of the Instrument Matrix

Finally, the instrument matrix can be assembled—the first three columnscan be drawn from the linear polarization-only measurements, while thelast column can be computed from {right arrow over (I)}_(RCP) and {rightarrow over (I)}_(LCP) cf. Eqn. 16.

A Consistency Check

Once the instrument matrix A has been determined, a consistency checkmay be conducted to ascertain whether it may be grossly inaccurate, akind of sanity check.

With all of the data from the calibration on hand, including from thefirst step with linear polarizations, the raw, un-normalized intensitydata recorded on each of the four channels observed as the linearpolarizer is rotated. For each linear polarizer orientation θ, recordingis made of an intensity vector {right arrow over (I)}_(θ) duringcalibration. Computation is performed of {right arrow over(S)}_(θ)=A⁻¹{right arrow over (I)}_(θ) which should ideally be a linearpolarization state with DOP=1.

In FIG. 17 , we plot this calculated DOP of the linear polarizationstates used during calibration. It can be observed that they neverdiffer by more than about 0.5% from 1. This means that the two parts ofthe calibration—with the linear polarizer alone and with the linearpolarizer and quarter-waveplate together—were in some senseself-consistent. If the DOP were not physical, one may not be able tohave faith in the obtained instrument matrix A.

Production of Partially Polarized Light Conceptual Foundation

Full-Stokes polarimetry, as its name connotes, provides for thedetermination of the entire Stokes vector. Notably, this allows for theanalysis of partially polarized beams of light for which the degree ofpolarization p may be less than unity.

Partially polarized light may be broken into two broad categories [54]:

Coherently depolarized, in which a beam is composed of many frequencies,each of which has its own time-independent Jones vector. If the Jonesvectors of all frequency components align, p=1, but if they aredifferent, in general p<1.

Incoherently depolarized, in which a beam, even if composed of onefrequency, may have p<1 if the parameters of the Jones vectors vary intime.

In our experiment, partially polarized light is produceddeterministically using a two polarization beamsplitters in aMach-Zehnder configuration [57], as in the text. The mechanism ofdepolarization here falls into the latter category: The beam (laserlight at λ=532 nm) is to a very good approximation monochromatic. Byvarying the distribution of power in the two arms of the interferometer,the light is incoherently depolarized by combining two beams with a lackof phase memory of one another. This configuration is corresponding to adevice used in fiber optics known as an air-gap polarization-dependentdelay line [54].

One important consideration in constructing this interferometer is thatthe path length difference be longer than the temporal coherence lengthL_(coh) of the source being used, so that the arms may truly lack phasecoherence with one another. Our laser source was a relativelyinexpensive diode-pumped solid state (DPSS) laser at λ=532 nm. In orderto test L_(coh) light is passed from the source into a simple Michelsoninterferometer, and varied the path length difference between arms untilinterference fringes were no longer visible [58]. From this quick,qualitative measurement, it is concluded that L_(coh) was on the orderof ˜1 cm.

As described in the present disclosure and depicted in FIG. 18 , as alinear polarizer is rotated in front of the first polarizationbeamsplitter, the amount of light distributed among the arms of theMach-Zehnder interferometer is varied and the degree of polarizationvaries from p=0 to 1 as dictated by the functional relation |cos 2θ|where θ is the angle of the linear polarizer with respect to the axis ofthe polarization beamsplitter.

As the linear polarizer is rotated in front of the Mach-Zehnderinterferometer, recording is made of the values on each of the fourphotodiodes behind the metasurface grating polarimeter. Using theinstrument matrix A obtained above, one may determine the incidentStokes vector {right arrow over (S)}_(inc) and thus its degree ofpolarization p. In the text, it is shown that as the angle of the linearpolarizer is varied from 0° to 90°, the measured DOP ranges from unityto a value of 1.2% at 45°.

Since

${p = \frac{\sqrt{S_{1}^{2} + S_{2}^{2} + S_{3}^{2}}}{S_{0}}},$the DOP contains error from every individual Stokes component. Measuringp near 0 presents an especially stringent specification; in order tomeasure p=0, the intensity readings on all channels should be equal(adjusting for differences in the efficiency of each channel). That is,the projection of unpolarized light onto any analyzer vector is equal.The low DOP reported in the present disclosure, then, speaks to theaccuracy of the metasurface grating polarimeter.

Experimental Comment

Described here some subtleties of this measurement as well asdifficulties encountered.

Full dataset: For the sake of completeness, it is noted that the partialDOP data presented in FIG. 3 d shows data for linear polarizerorientation between 0° and 90°, and a fraction of the data points takenin this range. Indeed, the LP angle changed in increments of 0.25° andthe measurement took place over the full range of LP angles between 0°and 360°. This full dataset is shown in FIG. 19 .

Sensitivity of alignment: Assurance is made that the front face of eachpolarization beam splitter (PBS) depicted in FIG. 18 was normal to theincident laser beam. In addition, great care was taken to align the twomirrors of the interferometer (both on tip/tilt mounts) so as to assurethat both arms combined into a single, co-linearly propagating outputbeam.

Beam divergence: Clearly, the beamsplitter configuration depicted inFIG. 18 has an optical path length difference between its arms; it isthis very effect that allows it to function as a depolarizer. However,due to a combination of the small size of the beam coming from thetelescope (it should be small enough to entirely fit inside the spatialextent of the 1.5 mm×1.5 mm grating structure) and imperfect beamre-collimation (the positions of the lenses are adjusted by hand), thebeam diverges by different amounts when traversing the two arms of theinterferometer. As a result of this effect, the beams coming from thetwo arms of the interferometer are not equal in size.

While this effect is minimized, it may not be truly eliminated. It istheorized that this effect is a major source of error in determining thedegree of polarization. In order to observe a DOP near zero, beams ofequal intensity from the two arms should interfere in order to producean incoherently depolarized beam. However, if the two beams possess adifferent spatial profile from one another, the degree of polarizationwill vary over the area of the resulting beam and the measured degree ofpolarization will be some spatial average that is, in general not 0.This effect is depicted in FIG. 21 . This could be a major contributorlimiting the minimum DOP measured.

When all of these effects were not fully accounted for, measurement ismade of some deviating results. Some of these are shown in FIG. 20 . Incertain cases (such as a), the DOP variation was not symmetric about45°. In other cases (b), the DOP follows the correct trend in angle butdoes not stay within the correct bounds. The DOP in b can be observedto, not return to the expected value of 1 while at other angles itexceeds it, which is unphysical. It is theorized that these erroneouscurves derive from a combination of the effects detailed above.

Error Propagation and Error Bars

It is aimed to be able to place uncertainty bounds on each element ofthe Stokes vectors computed with the meta-grating based polarimeter.Recall that the Stokes vectors are determined from the formula{right arrow over (S)}=A ⁻¹ {right arrow over (I)}

If the uncertainty on the elements of A and {right arrow over (I)} isknown, it is possible to propagate this uncertainty through Eq. 17 inorder to determine uncertainty bounds on all four Stokes coordinates.

First discussion is made of how uncertainty on A and {right arrow over(I)} is determined.

Uncertainty Bounds on Measured Intensities

Being a four-vector, the uncertainty in {right arrow over (I)} is mostgenerally described by a symmetric 4×4 covariance matrix which not onlycharacterizes the self-variance of each entry but also correlationsbetween variances of the elements.

In this disclosure, the intensity on each photodiode is sampled for aduration of ˜0.5 seconds. There is, then, a statistical distribution ofmeasured voltages during this time interval.

For each photodiode i, computation is made of the standard deviationσ_(i) of this distribution. This is taken to be the self variance ofchannel i. For simplicity, the covariances are regarded to be 0, anassumption which may well not be justified. The covariance matrix of{right arrow over (I)}, then, is taken to be diagonal. Except in thecase of, e.g., the laser source unexpectedly mode-hopping, variations in{right arrow over (I)} are generally very small; the error bars would,probably, not be much affected by neglecting the error in {right arrowover (I)} altogether.

Uncertainty in the Instrument Matrix

The instrument matrix A, though used to ascertain the Stokes vector{right arrow over (S)}, is itself also a measured quantity. The elementsof the 4×4 instrument matrix are themselves determined by thecalibration procedure. The matrix, being a list of 16 statisticalquantities, has an associated 16×16 symmetric covariance matrixquantifying its uncertainty.

The first three columns, that is, the leftmost 12 elements of theinstrument matrix, are determined from the calibration step involving alinear polarizer only. For each photodiode, the normalized intensity isrecorded as a function of the linear polarizer angle. Theoretically, theresulting data follows a three-parameter trigonometric curve for eachphotodiode. The first three elements of row i of the instrument matrixcome from the fitting parameters to the data from photodiode i.

The curves are fitted with a nonlinear least-squares regression which,along with the optimized fitting parameters, provides estimate of thecovariance matrix for the fitting parameters. The square root of thediagonal of this matrix provides the variance of each of the fittingparameters. Off-diagonal elements provide the covariance of the fittingparameters for photodiode i.

While we can obtain a covariance matrix for each of the individual curvefits, this technique provides no measure of the covariance betweenfitting parameters for different curves. Thus, covariances aredisregarded between different fitting parameters and the self-variance(e.g., the standard deviation) of each fitting parameter is kept.

The approximation is made that the covariance matrix for A, then, isdiagonal.

Next, uncertainty bounds are placed on the rightmost column of theinstrument matrix. This part is obtained from the step in thecalibration that uses both a quarter-waveplate and a linear polarizer.This procedure is done for both nominally LCP and RCP light, cf. FIG. 16. The difference in the mean values for LCP and RCP become the fourthcolumn of the instrument matrix. The variation of the voltage of eachphotodiode with angle is taken to be a statistical quantity, and takethe uncertainty in each element of the fourth column to be the standarddeviation of these variations in FIG. 16 . The final uncertainty in theelements of the fourth column is the difference between the mean valuesin both curves, so the errors from each curve have to be appropriatelypropagated. Notably here covariances are ignored—one could selectivelygenerate a covariance matrix for the fourth column, but there is noclear way to estimate the covariances of elements in the fourth columnwith the rest of the matrix. Moreover, this is likely not the bestestimate of the variance of these elements—the intensity variationsmanifest in FIG. 16 are in some sense expected, and are not a result ofrandom variations. Perhaps a better measure would be the deviation ofthe curve from a sine curve, its expected theoretical form.

In summary, the full 16×16 covariance matrix of the 4×4 instrumentmatrix A is assumed to be diagonal in form. This is perhaps a limitingassumption. Overall, however, this is intended to merely provideorder-of-magnitude estimates for the precision of the polarimeter.(Note: Once propagated through the matrix inversion process, thecovariance matrix of A⁻¹ is in general not diagonal [59]).

The above information is summarized in FIG. 22 .

Error in the Measured Stokes Vector

Obtained are estimates for the covariance cov(I_(k), I_(l)) between thekth and lth elements of {right arrow over (I)}, and the covariancecov(A_(ij), A_(qp)) between the ijth and qpth entries in A—both taken tobe diagonal matrices. It is desired to compute cov(S_(i), S_(j)), ormore generally the covariance matrix for the inferred Stokes vector.

An explicit solution to this problem in the context of polarimetry wasfirst presented in [59]. It is reproduced here.

The covariance matrix of {right arrow over (S)} is given bycov(S _(i) ,S _(j))=Σ_(α,β) I _(α) I _(β)cov(D _(iα) ,D _(iβ))+Σ_(k,l) D_(ik) I _(jl) cov(I _(k) ,I _(l))  (18)

The α, β summation is over all combinations of α=[1, 4] and β=[1, 4],with the same for the k, l summation. Here D=A−1. Its covariance matrixis given in the case of a square matrix by [59]cov(D _(αβ) ,D _(ab))=Σ_(i,j) D _(αi) D _(jβ) D _(αi) D _(jb)σ(A)_(ij)²  (19)where σ(A)_(ij) ² is the self-variance of the ijth element of A(assumption is made that A has a diagonal covariance matrix).

Error bars can be placed on the computed Stokes parameters by examiningthe diagonal elements of Eqn. 18. The full covariance matrix obtainedcan be used to propagate the error to quantities derived from the Stokesparameters.

Error in Derived Quantities

In the present disclosure, data is presented on several quantitiesderived from the Stokes vector {right arrow over (S)}. Presented here,for the sake of completeness, are error propagation formulae based onknowledge of the full covariance matrix of the Stokes vector.

Degree of polarization:

${p = \frac{\sqrt{S_{1}^{2} + S_{2}^{2} + S_{3}^{2}}}{S_{0}}},$so the variance in the degree of polarization

$\begin{matrix}{{\Delta\; p} = \sqrt{\begin{matrix}{{{\sigma^{2}\left( S_{0} \right)}\frac{S_{1}^{2} + S_{2}^{2} + S_{3}^{2}}{S_{0}^{4}}} + {\Sigma_{i = 1}^{3}\frac{{\sigma^{2}\left( S_{i} \right)}S_{i}^{2}}{S_{0}^{2}\left( {S_{1}^{2} + S_{2}^{2} + S_{3}^{2}} \right)}} -} \\{{\Sigma_{i = 1}^{3}\frac{2{{cov}\left( {S_{0},S_{i}} \right)}S_{i}}{S_{0}^{3}}} + {\Sigma_{i,j}\frac{2{{cov}\left( {S_{i},S_{j}} \right)}S_{i}S_{j}}{S_{0}^{2}\left( {S_{1}^{2} + S_{2}^{2} + S_{3}^{2}} \right)}}}\end{matrix}}} & (20)\end{matrix}$

The last sum is taken over all unique pairs i, j for values of i and jbetween 1 and 3 for which i≠j, σ²(S_(i))=cov(S_(i), S_(i)).

Azimuth: The azimuth θ of the polarization ellipse is given by

$\theta = {\frac{1}{2}\arctan{\frac{S_{2}}{S_{1}}.}}$Then the variance is obtained as

$\begin{matrix}{{\Delta\theta} = {\frac{1}{2}\sqrt{\frac{{S_{1}^{2}{\sigma^{2}\left( S_{2} \right)}} + {S_{2}^{2}{\sigma^{2}\left( S_{1} \right)}} - {2S_{1}S_{2}{{cov}\left( {S_{1},S_{2}} \right)}}}{S_{1}^{2} + S_{2}^{2}}}}} & (21)\end{matrix}$

Ellipticity: The ellipticity angle ϵ of the polarization ellipse isgiven by

$\in {= {\frac{1}{2}\arctan{\frac{S_{3}}{\sqrt{S_{1}^{2} + S_{2}^{2}}}.}}}$Its variance is obtained as

$\begin{matrix}{{\Delta \in} = {\frac{1}{2}\sqrt{\begin{matrix}{\frac{{\sigma^{2}\left( S_{3} \right)}\left( {S_{1}^{2} + S_{2}^{2}} \right)}{\left( {S_{1}^{2} + S_{2}^{2} + S_{3}^{2}} \right)^{2}} + \frac{{\sigma^{2}\left( S_{2} \right)}S_{2}^{2}S_{3}^{2}}{\left( {S_{1}^{2} + S_{2}^{2}} \right)\left( {S_{1}^{2} + S_{2}^{2} + S_{3}^{2}} \right)^{2}} +} \\{\frac{{\sigma^{2}\left( S_{1} \right)}S_{1}^{2}S_{3}^{2}}{\left( {S_{1}^{2} + S_{2}^{2}} \right)\left( {S_{1}^{2} + S_{2}^{2} + S_{3}^{2}} \right)^{2}} - \frac{{2{{cov}\left( {S_{1}S_{3}} \right)}S_{1}S_{3}} + {2{{cov}\left( {S_{2},S_{3}} \right)}S_{2}S_{3}}}{\left( {S_{1}^{2} + S_{2}^{2} + S_{3}^{2}} \right)^{2}}}\end{matrix}}}} & (22)\end{matrix}$

This expressions are derived straightforwardly from partial derivativesand error analysis. These values set the error bars on these quantitiesand utilize the full covariance matrix of the {right arrow over (S)}.

It should be noted that, by the very nature of these expressions, theerror is larger near some values than others. This can be seen in graphsof azimuth and ellipticity in the present disclosure. In regions whereS₁ is very nearly 0, the azimuth θ is very nearly 45°; the error bar onazimuth here will be larger due to dividing by a small quantity witherror. The same can be said of ellipticity near 45°.

Comparison with the Commercial Rotating Waveplate Polarimeter (RWP)General Description of Experiment

In the present disclosure, comparison is made the performance of themeta-grating based polarimeter to that of a commercial rotatingwaveplate polarimeter. In particular use is made of ThorLabs#PAX5710VIS-T, a visible frequency, free-space rotating waveplatepolarimeter.

In the experiment, laser light bounces off of two alignment mirrors,passes through a low-quality quarter-waveplate in order to compensatefor the laser's preferential polarization, and then goes through theaforementioned beam re-sizing optics to shrink the laser beam to a sizethat fits inside the 1.5 mm×1.5 mm metasurface grating. These featuresare common to FIGS. 23 and 24 . However, as is detailed below, the setupis changed slightly for different aspects of the comparison.

Comparison of Degree of Polarization

First, comparison is made of DOP readings between polarimeters. Asdepicted in FIG. 23 , the setup includes a rotating (Glan-Thompson)linear polarizer (LP) on a motorized rotation mount in front of thepreviously described Mach-Zehnder setup. The metasurface gratingpolarimeter (i) is placed in the beampath, and a list of random LPangular orientations is generated and stored. Intensity measurements arerecorded on all four photodiodes, and using the previously determinedinstrument matrix A, the resulting Stokes vector and its DOP arecomputed.

The metasurface grating polarimeter is replaced with the RWP (ii). Thesame LP angular orientations are visited, in the same order. The RWPdetermines the full-Stokes vector, including the measured DOP, and it isstored on a computer.

In the present disclosure, computation is made of the differences in theDOP measured by both polarimeters and find that the distribution ofdifferences has a mean value μ=0.6% and a standard deviation of σ=1.6%.Note that in the comparison plot, there is a bias toward high DOP due tothe shape of the |cos 2θ_(LP)| curve.

Comparison of Ellipticity and Azimuth

Next, comparison is made of the properties of the polarization ellipsereported by both polarimeters. Use is made of the setup in FIG. 24 ,which is mostly identical to that in FIG. 23 except for a zero-orderquarter-waveplate (QWP) on a motorized rotation mount in place of theMach-Zehnder interferometer. The metasurface grating polarimeter (i) isplaced in the beampath, and a set of random LP and QWP angularorientations is generated and stored. The list of orientations isre-ordered to optimize for travel time. Then, the LP and QWP visit eachof these angular orientations and the voltages on all four photodiodesare computed and converted to a Stokes vector by using A. Then,computation is made of the azimuth and ellipticity double angles:

$\begin{matrix}{{2\theta} = {\arctan\frac{S_{2}}{S_{1}}}} & (23) \\{and} & \; \\{{2\epsilon} = {\arctan\frac{S_{3}}{\sqrt{S_{1}^{2} + S_{2}^{2}}}}} & (24)\end{matrix}$

Consideration is made of 2θ and 2ϵ because these are the angularcoordinates from the Poincaré sphere, and are double the parameters ofthe actual ellipse.

The metasurface grating polarimeter is removed and the RWP is put in itsplace (ii). The same set of angles is revisited and the polarizationparameters reported by the RWP are stored on the computer.

In the present disclosure, computation is made of the differences in thethese quantities measured by both polarimeters and examine them asstatistical distributions. It is found that for azimuth 2θ, the meandifference is μ=0.004 rad and the standard deviation is σ=0.046 rad. Forellipticity 2ϵ, the mean difference is μ=0 rad and σ=0.015 rad. Thesequantities are multiplied by ½ for θ and ϵ alone.

Reason for Change in Setup

In the most general case, DOP, azimuth, and ellipticity can all varysimultaneously. Thus, for the most general comparison betweenpolarimeters, the LP/Mach-Zehnder configuration should be followed by asecond LP and a QWP. This way, by varying the angles of both LPs and theQWP, arbitrary states of polarization with varying overall intensity,DOP, azimuth, and ellipticity would be generated and compared.

It is noticed however, that the inclusion of the Mach-Zehnderinterferometer to generate arbitrary DOP added significant noise to thedetermined polarization ellipse. Even without touching anything, it isnoticed that the reading can vary significantly within the time scale ofa second. When the mirrors of the Mach-Zehnder were brushed lightlywith, e.g., a fingertip, the polarization ellipse would changedrastically.

This effect is noticed in the readings from both the metasurface gratingpolarimeter and the commercial RWP, and it was particularly pronouncedat low DOP. After all, the DOP p quantifies the ratio of the beam thatmay be considered fully polarized to that part that is totallyunpolarized. Consequently, when trying to determine the parameters ofthe polarization ellipse at low DOP, the power that informs such ameasurement is a small fraction of that of the incident beam. Comparedto the unpolarized background, this is consequently harder to measure,and it is also more affected by noise in the measurement.

Therefore, a comparison of fully arbitrary polarization states betweenthe two polarimeters would present an unfair assessment of theperformance of the metasurface grating polarimeter. Since thepolarization ellipse parameters were especially jittery on bothpolarimeters, this effect may be encountered twice over since the RWP isregarded to be an absolute polarization reference. Therefore, comparisonis made of DOP and ellipticity/azimuth from separate datasets. The fullparameters from both such datasets—with and without the Mach-Zehnder—arepresented in FIGS. 25 and 26 , respectively. The portions of thesefigures used in present disclosure FIG. 4 a are highlighted with adashed box. The standard deviations of azimuth/ellipticity in FIG. 25are indeed about double those in FIG. 26 , and the DOPs measured in FIG.26 are all about 1.0, as is to be expected.

Data Post-Processing

Documented here is post-processing performed on the polarimetercomparison dataset before it is presented in FIG. 4 of the presentdisclosure.

Azimuth offset: During calibration of the metasurface gratingpolarimeter, the orientation of the linear polarizer which is labeled“0” becomes the origin of the polarimeter's coordinate system.Similarly, when the RWP is calibrated, it too is imbued with acoordinate system. These coordinate systems are in general rotated by anangle relative to one another called the azimuth offset. This is aconstant offset to azimuth between data reported by the twopolarimeters. For the sake of simplicity, subtraction is made from allmetasurface azimuth data the mean difference in azimuths reported byboth polarimeters. This is perhaps not the best scheme—in essence, thismeans that the metasurface azimuth data is shifted so as to assure thatthe mean difference in azimuth values between the polarimeters is 0.Thus, the mean of the azimuth histogram has no meaning. A better,unbiased strategy would be to measure the state produced with the LP at0° with the RWP and use its azimuth as the azimuth offset. However, manycalibrations are run and this additional step is omitted. This may beemployed in other implementations presenting a thorough errorcharacterization of the metasurface grating polarimeter. It is note thatthis procedure has no effect on the comparison of the ellipticitybetween polarimeters.

Distinction between RCP and LCP: Which handedness of circularpolarization is called “right” or “left” is ultimately a subjectivedistinction. Which the metasurface grating polarimeter “calls” right (orleft) is set during the step of the calibration involving a LP and QWPtogether; whichever is deemed “right” there becomes right for thepolarimeter. This convention may differ between the two polarimeters,which would manifest itself as a negative ellipticity correlation ratherthan positive. If this is present, the trend of the data is reversed tocorrect for the arbitrary distinction.

Pruning of data with especially high error bars: A peculiarity of the(very basic, inexpensive) λ=532 nm laser source used in our experimentis that it tends to occasionally hop modes, producing sharp fluctuationsin output power. Measurements by the metasurface grating polarimeterwere averaged over a 0.5 s integration time, and the resultingpolarimetric measurement is skewed if this occurs. Since themeasurements were completely automated, there was no means ofidentifying this at-the-time-of-measurement to discard these datapoints. Instead, these points are identifiable by their especially largeerror bars. Fluctuation during measurement produces high error in {rightarrow over (I)}which is propagated to measured quantities. Inparticular, removal is made of any points from the data whose error inany of the three quantities (DOP, azimuth, ellipticity) is larger thanthree times the mean value of error for the dataset.

Comment on Results

What exactly do the histograms presented in present disclosure FIG. 4 amean? It is the error bars on the individual data points which quantifythe precision of the measurements (e.g., the range of measured valuesexpected over many subsequent iterations of the measurement with thesame incident polarization). However, in certain regimes one may expectthat these error bars are larger than others (as discussed above); thatis, the size of the error bar for, e.g., azimuth or ellipticity is inpart a function of the azimuth or ellipticity itself. In the histograms,the statistical distribution of the difference is considered between themeasured value and (what is taken to be) the true value. Depending onthe polarization states of the test points, these distributions could beartificially broadened or narrowed. That is if, e.g., we concentrate onpolarization states whose azimuths fall in the regions where theprecision on azimuth is consequently higher, the distribution will benarrower. If measurements are concentrated in a region where, e.g., theprecision of azimuth is less, the distribution will be artificiallybroadened, and one would say that the polarimeter is less accurate. Inorder to be unbiased, the measurement points should be uniformlydistributed in the entire range of the given parameter, so that all ofthe best cases and the worst cases are uniformly represented. If this isdone, the standard deviation of the distribution of differences betweenmeasurement and a reference can be indicative of the accuracy of thepolarimeter in that particular quantity; the mean is indicative of somesystematic error that causes the polarimeter to be off by a constantamount relative to a reference.

TABLE I Comparison of accuracy results from the polarimeter comparisonMetagrating ThorLabs Spec Sheet Quantity Polarimeter (RWP) Degree ofPolarization 1.6% 0.5% (DOP) Azimuth 1.32° 0.2° Ellipticity 0.43° 0.2°

In this disclosure, exploration is made of (somewhat uniformly) allpossible values of DOP, azimuth, and ellipticity relative to what istreated as an absolute polarization reference, that being the RWP. To agood approximation, then, one can treat the σ of these differences to bea measure of the accuracy.

The results of this are listed in Table I, with values for the RWPobtained from the spec sheet (as of this time listed on ThorLabs'webpage. In comparing the values, one should keep in mind that how theaccuracies of the RWP are obtained are unknown, so whether they aredirectly comparable is unknown. Additionally, the RWP is treated as anabsolute polarization reference. If this is not the case (it iscertainly not), it will increase the perceived error of the metasurfacegrating polarimeter.

The overall point made here is that the accuracies for the metasurfacegrating polarimeter are in the same ballpark as that of the RWP, despitebeing a device with no moving parts or bulk polarization optics. Withfurther optimization and engineering, the two polarimeters couldunquestionably become comparable.

Angle Dependence of Polarization Production and Polarimetry

A practical question that arises with any new diffractive opticalelement is that of angle-dependence. In this disclosure, metasurfacepolarization gratings are presented that may generate specific states ofpolarization when a known polarization state is incident. Additionally,the grating may function as a polarimeter. In this section, examinationis made of angle-of-incidence effects on both of these applications.

In the strictest sense, the phase shift produced by the individualmetasurface subwavelength elements is valid for a given angle. However,as will be seen in what follows, the metasurface is surprisingly robustwith respect to angle.

Effect of Off-Normal Incidence on the Produced Polarization Ellipses

The metasurface is designed so that, when illuminated with 45° linearlypolarized light, desired polarization states are produced on thediffraction orders. In order to investigate this effect, the experimentshown schematically in FIG. 27 is performed. Light passes through alinear polarizer (LP) oriented at 45° relative to the coordinate systemof the diffraction grating. The grating is turned about its center at anangle of θ relative to the beam, and the polarization ellipses of thelight on the diffraction orders are measured with a commercial RWP.

This measurement is carried out for θ between −40° and +40°, forgratings producing a tetrahedron of polarization ellipses and thegrating producing +45°/RCP/LCP/−45°. The results are presented in avariety of forms in FIGS. 28, 29, and 30 .

Effect of Off-Normal Incidence on Polarimeter Calibration and StokesVector Determination

Description of Error Analysis Scheme

A polarimeter is calibrated with a beam impinging at one angle ofincidence; if it is then used at a different angle of incidence, thedetermined Stokes vector will consequently contain error. This is aneffect relevant to all polarimeters, including the rotating waveplatepolarimeter where the angle of incidence on the rotating QWPunquestionably affects the retardance observed.

A small study is conducted on the metasurface grating polarimeter todetermine its susceptibility to angle dependent effects. If thepolarimeter is calibrated for a 0° angle-of-incidence, how much could anend user accidentally tilt the device and still expect to obtainreasonably accurate results?

A scenario is considered in which the polarimeter is used assuming aninstrument matrix obtained from a calibration at, say, 0° incidence, ata different angle. In this case then, there is a “perceived” instrumentmatrix A_(p) which the user applies believing that is correct, and an“actual” instrument matrix A_(a) that reflects the true behavior of thepolarimeter. If the user makes an observation in the form of anintensity vector {right arrow over (I)}_(meas), they will report ameasured Stokes vector {right arrow over (S)}_(p)=A_(p) ⁻¹{right arrowover (I)}_(meas). The Stokes vector in actuality, however, is {rightarrow over (S)}_(a)=A_(a) ⁻¹{right arrow over (I)}_(meas), meaning thatthere is a measurement error which one can express in vector form:{right arrow over (Δ)}({right arrow over (I)} _(meas))=(A _(p) ⁻¹ −A_(a) ⁻¹){right arrow over (I)} _(meas)  (25)

Notably, the error {right arrow over (Δ)} is itself a function of themeasured intensities {right arrow over (I)}_(meas). Those measuredintensities are a function of the actual incident polarization state,{right arrow over (S)}_(a). Thus, the error accrued from use of theincorrect instrument matrix (because of e.g., an angular tilt of thepolarimeter) has a polarization dependence.

The above discussion suggests a Monte-Carlo-like error analysis scheme.This scheme is depicted in FIG. 26 . One can start with a set of actualStokes vectors that might be incident on the polarimeter, {{right arrowover (S)}_(a)}. Using the actual instrument matrix of the device (e.g.,the instrument matrix at a given tilt θ), computation can be made, foreach of the actual Stokes vectors of a set of measured intensity vectorsthat would actually be observed by the user, {{right arrow over(I)}_(meas)}. Since the user makes use of an incorrect, perceivedinstrument matrix A_(p), when these measured intensities are analyzed,they will compute a set of perceived Stokes vectors {{right arrow over(S)}_(p)}=A_(p) ⁻¹{{right arrow over (I)}_(meas)}. Of course, ifA_(p)=A_(a), the user will reproduce the actual Stokes vectors, but ingeneral this is not the case, and the error of the polarimeter can beanalyzed by comparing the elements of {{right arrow over (S)}_(a)} tothose of {{right arrow over (S)}_(p)} (FIG. 26 ). There are manypossible metrics for quantifying this error.

This error analysis scheme shows how a given Stokes vector will bemeasured by a polarimeter that is out of its calibration condition. Ingeneral, the whole Stokes vector will be deformed, so a user measuring agiven beam will not only observe a different polarization ellipse, butmay also report a different beam power or a beam that appearsartificially polarized or depolarized.

Incident Angle Error Analysis of Meta-Grating Polarimeter

The difference between A_(a) and A_(p) could come from any error source,but in the present case it is desired to examine the effect of angle ofincidence. To this end, calibration is made of the polarimeter using themethods extensively detailed in this disclosure at normal incidence.Then, the calibration is conducted at several other angles of incidence(the meta-grating is mounted on a rotation mount, akin to the experimentshown in FIG. 27 ). In particular, the calibration is conducted atincident angles of 5°, 10°, 15°, and 20°.

In a realistic situation, a user will believe that they are using theinstrument matrix from calibration, that is, A_(p)=A_(0°). If, however,the device is tilted at angle θ, A_(a)=A_(θ). The analysis detailedabove is performed for all incident angles for which calibration is madeof the polarimeter. Consideration is made of the simulated inputpolarizations {{right arrow over (S)}_(a)} a uniform sampling of pointson the surface of the Poincare sphere. (In a more general analysis, theentire volume of the sphere would be included too to account forpartially polarized inputs).

The analysis described above is conducted; the results are depicted inFIG. 32 a-c . One starts with a set of incident Stokes vectors {{rightarrow over (S)}_(a)} which more or less sample the surface of thePoincare sphere (left side of FIG. 32 a-c ). Then, at each angle θ, the“perceived” Stokes vectors {{right arrow over (S)}_(p)} that would bereported are computed using the scheme in FIG. 31 . The set {{rightarrow over (S)}_(p)} for each angle is plotted on the Poincare sphere inFIG. 32 a-c , and its error relative to the set {{right arrow over(S)}_(a)} is evaluated using several metrics. One of these is theabsolute error in degree of polarization (DOP) between {{right arrowover (S)}_(a)} and {{right arrow over (S)}_(p)}. The results for eachangle, including the maximum and minimum DOP error for the entire set,are displayed in FIG. 32 a-c , where the error in DOP is depicted by acolor scale between green and red with red indicating the highest error.

The Stokes vectors constituting the rows of the instrument matrix arefrequently referred to as analyzer vectors because any incident Stokesvector is projected on these to produce the measured intensities {rightarrow over (I)}. Also shown in FIG. 32 a-c are the analyzer vectors atangle θ (the rows of the instrument matrix A_(θ)) alongside the idealtetrahedron.

Discussion/Conclusion

From the previous study whose results are detailed in Table I, theuncertainty on the DOP, azimuth, and ellipticity of measuredpolarization ellipses are known for normal-incidence polarimetry. Inorder to evaluate the effect of off-normal incidence on polarimetricperformance, the σ of the errors for each quantity (displayed in FIG. 32a-c ) may be compared to the precision (in Table III). Even at a 5°misalignment, the errors exceed the uncertainties measured at normalincidence, but not exceedingly so (6% vs. 1.6% for DOP, 2.96° vs. 1.32°for azimuth, and 1.59° vs. 0.43° for ellipticity). Qualitatively, then,the polarimeter could suffer an accidental misalignment of 5° or lessand still offer reasonable accuracy, with some polarization states thatwould yield particularly high error (clustered around the red regions inFIG. 32 a-c ).

While the present disclosure has been described and illustrated withreference to specific embodiments thereof, these descriptions andillustrations do not limit the present disclosure. It should beunderstood by those skilled in the art that various changes may be madeand equivalents may be substituted without departing from the truespirit and scope of the present disclosure as defined by the appendedclaims. The illustrations may not be necessarily drawn to scale. Theremay be distinctions between the artistic renditions in the presentdisclosure and the actual apparatus due to manufacturing processes andtolerances. There may be other embodiments of the present disclosurewhich are not specifically illustrated. The specification and drawingsare to be regarded as illustrative rather than restrictive.Modifications may be made to adapt a particular situation, material,composition of matter, method, or process to the objective, spirit andscope of the present disclosure. All such modifications are intended tobe within the scope of the claims appended hereto. While the methodsdisclosed herein have been described with reference to particularoperations performed in a particular order, it will be understood thatthese operations may be combined, sub-divided, or re-ordered to form anequivalent method without departing from the teachings of the presentdisclosure. Accordingly, unless specifically indicated herein, the orderand grouping of the operations are not limitations of the presentdisclosure.

All of the publications, patent applications and patents cited in thisspecification are incorporated herein by reference in their entirety.

What is claimed is:
 1. A detector comprising: a substrate; an array ofsubwavelength-spaced phase-shifting elements disposed on the substrateand configured to produce, when illuminated with a polarized incidentlight, a diffracted light beam with a distinct polarization state foreach of a finite number of diffraction orders, wherein the finite numberis 2 or more; and one or more detecting elements, each configured todetect the diffracted light beam for one of the diffraction orders. 2.The detector of claim 1, wherein the array is a one-dimensional array.3. The detector of claim 1, wherein the phase-shifting elements comprisepillars.
 4. The detector of claim 3, wherein the pillars have arectangular cross-section.
 5. The detector of claim 1, wherein thephase-shifting elements comprise titanium dioxide, silicon nitride, anoxide, a nitride, a sulfide, or a pure element.
 6. The detector of claim1, wherein the subwavelength-spaced phase-shifting elements aretessellated on the substrate and are configured to produce, whenilluminated with the polarized incident light, four distinctpolarization states.
 7. The detector of claim 6, wherein thesubwavelength-spaced phase-shifting elements are further configured toproduce a +45° linear polarization state, a right circular polarizationstate, a left circular polarization state and a −45° linear polarizationstate at −2, −1, +1 and +2 diffraction orders, respectively, whenilluminated with the polarized incident light which is +45° linearpolarized relative to a surface of the substrate.
 8. The detector ofclaim 6, wherein the subwavelength-spaced phase-shifting elements arefurther configured to produce at −2, −1, +1 and +2 diffraction ordersfour polarization states corresponding to vertices of a tetrahedroninscribed in a Poincaré sphere when illuminated with the polarizedincident light which is +45° linear polarized relative to a surface ofthe substrate.
 9. The detector of claim 1, wherein each dimension of thesubstrate parallel to a surface of the substrate is no more than 2millimeters.
 10. The detector of claim 1, wherein a number of thesubwavelength-spaced phase-shifting elements is in a range from 10 to40.
 11. The detector of claim 1, wherein polarization states for each ofthe diffraction orders are linearly independent.
 12. The detector ofclaim 1, wherein the one or more detecting elements comprise one or moresingle-wavelength detecting elements, one or more multi-wavelengthdetecting elements, or one or more imaging sensors.
 13. The detector ofclaim 12, wherein the single-wavelength detecting elements are linear inintensity at a wavelength of the polarized incident light.
 14. Thedetector of claim 1, further comprising a first linear polarizerpositioned on an optical path of an incident light towards the array ofsubwavelength-spaced phase-shifting elements.
 15. The detector of claim14, further comprising a second linear polarizer on an optical path ofthe diffracted light beam towards the one or more detecting elements.16. The detector of claim 15, wherein an extinction ratio of the secondlinear polarizer is in a range from 500 to
 200000. 17. The detector ofclaim 1, further comprising a lens positioned on an optical path of thediffracted light beam towards the one or more detecting elements. 18.The detector of claim 1, wherein the one or more detecting elementscomprises one or more imaging sensor elements.
 19. A polarization testmethod comprising: providing a detector comprising: a substrate; anarray of subwavelength-spaced phase-shifting elements disposed on thesubstrate and configured to produce, when illuminated with a polarizedincident light, a diffracted light beam with a distinct polarizationstate for each of a finite number of diffraction orders, wherein thefinite number is 2 or more; and one or more detecting elements, eachconfigured to detect the diffracted light beam for one of thediffraction orders; illuminating the array of subwavelength-spacedphase-shifting elements with a test light; and measuring a lightintensity of a beam diffracted from the array of subwavelength-spacedphase-shifting elements for each of the finite number of the diffractionorders using the one or more detecting elements.
 20. The method of claim19, wherein the test light is a partially polarized or an unpolarizedlight.
 21. The method of claim 19, wherein an incident angle of the testlight differs from a calibration incident angle of the optical componentby ±5°.